Radiative decay of the X(3872) state as a mixture of molecule and charmonium

Nuclear Physics B (Proc. Suppl.) 207–208 (2010) 253–256 www.elsevier.com/locate/npbps

Radiative decay of the X(3872) state as a mixture of molecule and charmonium C. M. Zanettia,∗, M. Nielsena a Instituto

de F´ısica, Universidade de S˜ao Paulo, C.P. 66318, 05389-970 S˜ao Paulo, SP, Brazil

Abstract We use QCD sum rules to calculate the width of the radiative decay of the meson X(3872), assumed to be a mixture between charmonium and exotic molecular [cq][q¯ ¯ c] states with J PC = 1++ . We find that in a small range for the values 0 0 of the mixing angle, 5 ≤ θ ≤ 13 , we get the branching ratio Γ(X → J/ψγ)/Γ(X → J/ψπ+ π− ) = 0.19 ± 0.13, which is in agreement, with the experimental value. This result is compatible with the analysis of the mass and decay width of the mode J/ψ(nπ) performed in the same approach. 1. Introduction The X(3872) state has been first observed by the Belle collaboration in the decay B+ → X(3872)K + → J/ψπ+ π− K + [1], and was later confirmed by CDF, D0 and BaBar [2]. The current world average mass is mX = (3871.4±0.6) MeV, and the width is Γ < 2.3 MeV at 90% confidence level. Babar collaborations reported the radiative decay mode X(3872) → γJ/ψ [3, 4], which determines C = +. Belle Collaboration reported the branching ratio: Γ(X → J/ψγ) = 0.14 ± 0.05. Γ(X → J/ψ π+ π− )

(1)

Further studies from Belle and CDF that combine angular information and kinematic properties of the π+ π− pair, strongly favors the quantum numbers J PC = 1++ or 2−+ [3, 5, 6]. The interest in this new state has been increasing, since the mass of the X(3872) could not be related to any charmonium state with the quantum numbers J PC = 1++ in the constituent quark models [7], indicating that the conventional quark-antiquark structure should by abandoned in this case. Another interesting experimental finding is the fact that the decay rates of the processes X(3872) → J/ψ π+ π− π0 and X(3872) → J/ψπ+ π− are comparable [3], indicating a strong isospin and G parity violation, which is incompatible with a c¯c structure for X(3872). ∗ Speaker

Email addresses: [email protected] (C. M. Zanetti), [email protected] (M. Nielsen)

0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2010.10.065

The isospin violation problem can be easily avoided in a multiquark approach. In this context the molecular picture has gained attention. The observation of the above mentioned decays, plus the coincidence between the X mass and the D∗0 D0 threshold: M(D∗0 D0 ) = (3871.81 ± 0.36) MeV [8], inspired the proposal that the X(3872) could be a molecular (D∗0 D¯ 0 − D¯ ∗0 D0 ) bound state with small binding energy [9, 10]. There are also some experimental data that seem to indicate the existence of a c¯c component in the X(3872) structure [11, 13]. However, as pointed out in ref. [14], a consistent analysis of the D0 D¯ ∗0 molecule production requires taking into account the effect of final state interactions of the D and D∗ mesons. Besides this debate, the recent observation, reported by BaBar [15] of the decay X(3872) → ψ(2S )γ is much bigger than the molecular prediction [16]. In Ref.[17] the QCDSR approach was used to study the X structure including the possibility of the mixing between two and four-quark states. This was implemented following the prescription suggested in [18] for the light sector. The mixing is done at the level of the currents and is extended to the charm sector. In a different context (not in QCDSR), a similar mixing was suggested already some time ago by Suzuki [13]. Physically, this corresponds to a fluctuation of the cc state where a gluon is emitted and subsequently splits into a light quark-antiquark pair, which lives for some time and behaves like a molecule-like state. The possibility that the X(3872) is the mixing of two-quarks and fourquarks was also considered to investigate the radiative decay in the effective Lagrangian approach [19].

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In this work we will focus on the radiative decay X(3872) → J/ψγ. We use the mixed two-quark and four-quark prescription of Ref.[17] to perform a QCD sum rule analysis of the radiative decay X(3872) → J/ψ γ.

where p = p + q and the interpolating fields are given by: jψμ = c¯a γμ ca ,  jγν = eq qγ ¯ νq ,

(9) (10)

q=u,d,c

2. The mixed two-quark / four quark operator The mixed charmonium-molecular current proposed in Ref.[17] will be used to study radiative decay of the X(3872) in the QCD sum rules framework. For the charmonium part we use the conventional axial current: j(2) ¯ a (x)γμ γ5 ca (x). μ (x) = c

(2)

The D D∗ molecule is interpolated by [20–22]:   1  (4q) q¯ a (x)γ5 ca (x)¯cb (x)γμ qb (x) jμ (x) = √ 2   (3) − q¯ a (x)γμ ca (x)¯cb (x)γ5 qb (x) ,

with eq = 23 e for quarks u and c, and eq = − 31 e for quark d (e is the modulus of the electron charge). The current JμX is given by the mixed charmonium-molecule current in Eq. (6). In our analysis, we consider the quarks u and ¯ = d to be degenerate, i.e., mu = md and ¯uu = dd qq. ¯ To evaluate the phenomenological side of the sum rule we insert, in Eq.(7), the intermediate states for X and J/ψ, obtaining the following equation: Πμνα (p, p , q) = phen

 ×

As in Ref. [18] we define the normalized two-quark current as j(2q) = μ

1 √ ¯uu j(2) μ , 6 2

(4)

and from these two currents we build the following mixed charmonium-molecular current for the X(3872): Jμq (x)

=

sin(θ) j(4q) μ (x)

+

cos(θ) j(2q) μ (x).

jμX (x) = cos αJμu (x) + sin αJμd (x),

(6)

Jμq (x),

with (q = u, d), given by the mixed twoquark/four-quark current in Eq. (5). 3. The three point correlator In this section we use QCD sum rules [23, 24]to study the vertex associated to the decay X(3872) → J/ψγ. The QCD sum rule calculation for the vertex X(3872) J/ψ γ is centered around the three-point function given by    Πμνα (p, p , q) = d4 xd 4 y eip .x eiq.y Πμνα (x, y), (7) with †

Πμνα (x, y) = 0|T [ jψμ (x) jγν (y) jαX (0)]|0,

(8)

m2X (p2 − m2X )(p 2 − mψ )

αμνσ qσ p · q A + μνλσ pλ qσ qα B

−

ανλσ qμ qσ pλC + ανλσ pλ pμ qσ (C − A)

−

μνλσ pλ qσ (qα + pα )(A + B)

p·q , m2X

p·q m2ψ (11)

where we have used the matrix element that describes the decay X → γJ/ψ given by [19]: μ

ρ

M(X(p) → γ(q)J/ψ(p )) = e εκλρσ Xα (p) ψ (p ) γ (q)×

(5)

Following Ref. [17] we will consider a D0 D¯ ∗0 molecular state with a small admixture of D+ D∗− and D− D∗+ components:

ie(cos α + sin α)λq mψ fψ

×

qσ (A gμλ gακ p · q + Bgμλ pκ qα + Cgακ pλ qμ ), (12) m2X

where A, B, C are dimensionless couplings. We also use the following definitions: 0| jψμ |ψ(p ) = mψ fψ μ (p ) ; X(p)| jαX |0 = (cos α + sin α)λq α∗ (p) ,

(13) (14)

where the meson-current coupling parameter is extracted from the two-point function, and its value was obtained in Ref. [17]: λq = (3.6 ± 0.9) × 10−3 GeV5 . In the OPE side we work in leading order in α s and we consider condensates up to dimension five. In the phenomenological side, as we can see in Eq. (11), there are five independent structures. Taking the limit p2 = p 2 = −P2 and doing a single Borel transform to P2 → M 2 , we arrive at a general formula for the sum rule for each structure i, after matching the OPE and the phenomenological side:  2 2  2 2 2 Gi (Q2 ) e−mψ /M − e−mX /M + Hi (Q2 ) e−s0 /M = ¯ (OPE) (M 2 , Q2 ), =Π i

(15)

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0.060 0.055 0.050

0.008

0.014

0.007

0.012

0.006

0.010

0.030

3

0.004

0.008

3

2

0.035

0.005

-G (GeV )

0.040

-G (GeV)

G1(GeV3)

0.045

0.006

0.003 0.004

0.025

0.002

0.020 0.0

0.5

1.0

1.5

2.0

2.5 2

3.0

3.5

4.0

4.5

0.001

0.002 0.0

0.5

1.0

2

1.5

2.0

2.5 2

Q (GeV )

3.0

3.5

4.0

4.5

0.0

0.5

1.0

2

1.5

2.0

2.5 2

Q (GeV )

3.0

3.5

4.0

4.5

2

Q (GeV )

Figure 1: Momentum dependence of the functions for s1/2 = 4.4 GeV and u1/2 = 3.6 GeV: G1 , G2 and G3 . The solid line gives the parametrization of the QCDSR 0 0 results (dots) through Eq. (19) and the results in Table 1.

where Q2 = −q2 and Hi (Q2 ) gives the contribution of the pole-continuum transitions [25–27]. The RHS of Eq. (15) is the OPE side evaluated from Eq. (7). In the LHS, since the photon is off-shell in the vertex XJ/ψγ, instead of the couplings constants A, B and C we describe the vertex in terms of form factors. The functions Gi (Q2 ) in Eq. (15) are directly proportional to the form factors A(Q2 ), B(Q2 ) and C(Q2 ). We choose the structures αμνσ qσ , μνσλ pσ pα qλ and μνσλ pσ pα qλ , to determine the form factors. 4. Numerical analysis The sum rules are analyzed numerically using the following values for quark masses and QCD condensates [28, 29], and for meson masses e decay constants:

should not depend on M 2 , so we limit our fit to regions where the functions are clearly stable in M 2 to all values of Q2 . The regions of stability in M 2 for G1 (Q2 ) is 7.0 GeV2 ≤ M 2 ≤ 8.5 GeV2 , for G2 (Q2 ) is 6.5 GeV2 ≤ M 2 ≤ 7.5 GeV2 , and for G3 (Q2 ) is 8.0 GeV2 ≤ M 2 ≤ 9.0 GeV2 . The form factors A(Q2 ), (A(Q2 ) + B(Q2 )), and C(Q2 ) are obtained respectively from the functions on the LHS G1 (Q2 ), G2 (Q2 ) and G3 (Q2 ). Since the coupling constants, appearing in Eq. (12), are defined as the value of the form factors at the photon pole, Q2 = 0, to determine the couplings A, B and C we have to extrapolate A(Q2 ), B(Q2), and C(Q2 ) to a region where the sum rules are no longer valid (since the QCDSR results are valid at the deep Euclidean region). To do that we fit the QCDSR results, shown in Fig. (1), as exponential functions:

mc (mc ) = (1.23 ± 0.05) GeV,

Gi (Q2 ) = g1 e−g2 Q . 2

qq ¯ = −(0.23 ± 0.03)3 GeV3 , m20 = 0.8 GeV2 , mψ = 3.1 GeV mX = 3.87 GeV,

fψ = 0.405 GeV

(16)

The value of the angle α that defines the mixing between the D0 D¯ ∗0 , D¯ 0 D∗0 and D+ D∗− , D− D∗+ has been obtained previously in Ref. [17, 25, 30]:

(19)

We do the fitting for s1/2 = 4.4 GeV and u1/2 = 0 0 3.6 GeV as the results do not depend much on this parameters. The numerical values of the fitting parameters are shown in the Table 1. Table 1: Results for the fitting parameters.

α = 20o

(17)

For the mixing angle of two and four quark states, θ, we use the values that were obtained in the QCD sum rules analysis of the mass of the X and the decay mode X → J/ψ(nπ) [17]: θ = (9 ± 4)o .

(18)

In the LHS of Eq. (15), the unknown functions Gi (Q2 ) and Hi (Q2 ) have to be determined by matching both sides of the sum rule. The functions Gi (Q2 )

g1 g2

G1 0.056 GeV3 0.25 GeV−2

G2 −0.0069 GeV 0.365 GeV−2

G3 −0.013 GeV3 0.41 GeV−2

From Fig. (1) we can see that the Q2 dependence of the QCDSR results for the functions Gi (Q2 ) are well reproduced by the chosen parametrization, in the interval 2.5 GeV2 ≤ Q2 ≤ 4.5 GeV2 , where the QCDSR are valid. With the results for the functions on the LHS of the sum rules (15), we compute the coupling constants.

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Varying θ in the range 5o ≤ θ ≤ 13o we get the following results:

A = A+B = C

A(Q2 = 0) = 18.65 ± 0.94 ; (A + B)(Q2 = 0) = −0.24 ± 0.11 ;

= C(Q2 = 0) = −0.843 ± 0.008 .

(20)

The decay width is given in terms of these couplings through: Γ(X → J/ψ γ) =

 m2X α p∗5 2 2 (A+B) , (21) + (A+C) 3 m4X m2ψ

where p∗ = (m2X − m2ψ )/(2mX ) is the three-momentum of the decay products. To compare our results with the experimental data shown in Eq. (1) we use the result for the decay width of the channel J/ψπ+ π− , obtained in the Ref. [17], which was computed in the same range of the mixing angle θ: Γ(X → J/ψ ππ) = 9.3 ± 6.9 MeV. We get Γ(X → J/ψ γ) = 0.19 ± 0.13 , Γ(X → J/ψ π+ π− )

(22)

which is in complete agreement with the experimental result. 5. Conclusions We have presented a QCDSR analysis of the threepoint function of the radiative decay of the X(3872) meson by considering a mixed charmonium-molecular current. We find that the sum rules results in Eqs. (22) are compatible with experimental data. These results were obtained by considering the mixing angles in Eq. (6) and (5) with the values α = 20o and 5◦ ≤ θ ≤ 13◦ . The present result is also compatible with previous analysis of the mass of the X state and the decays into J/ψπ0 π+ π− and J/ψπ+ π− [17], since the values of the mixing angles used in both calculations are the same. It is important to mention that there is no free parameter in the present analysis and, therefore, the result presented here strengthens the conclusion reached in Ref. [17] that the X(3872) is probably a mixed state of charmonium and molecule. Acknowledgements This work has been supported by FAPESP.

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