Radiative decay of the X(3872) as a mixed molecule-charmonium state in QCD Sum Rules

# Radiative decay of the X(3872) as a mixed molecule-charmonium state in QCD Sum Rules

M. Nielsen Instituto de Física, Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, SP, Brazil    C.M. Zanetti Instituto de Física, Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, SP, Brazil
###### Abstract

We use QCD sum rules to calculate the width of the radiative decay of the meson , assumed to be a mixture between charmonium and exotic molecular states with . We find that in a small range for the values of the mixing angle, , we get the branching ratio , which is in agreement, with the experimental value. This result is compatible with the analysis of the mass and decay width of the mode performed in the same approach.

###### pacs:
11.55.Hx, 12.38.Lg , 12.39.-x

## I Introduction

The state has been first observed by the Belle collaboration in the decay belle1 (), and was later confirmed by CDF, D0 and BaBar Xexpts (). The current world average mass is , and the width is at 90% confidence level. Babar collaborations reported the radiative decay mode belleE (); babar2 (), which determines . Belle Collaboration reported the branching ratio:

 Γ(X→J/ψγ)Γ(X→J/ψπ+π−)=0.14±0.05. (1)

Further studies from Belle and CDF that combine angular information and kinematic properties of the pair, strongly favors the quantum numbers or belleE (); cdf2 (); cdf3 (). Between these quantum numbers, a recent BaBar measurement favors the assignment babarate (). However, established properties of the are in conflict with this assignment kane (); bpps () and, therefore, in this work we assume the quantum numbers of the to be .

The interest in this new state has been increasing, since the mass of the could not be related to any charmonium state with the quantum numbers in the constituent quark models bg (), 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 and are comparable belleE ():

 Γ(X→J/ψπ+π−π0)Γ(X→J/ψπ+π−)=1.0±0.4±0.3. (2)

This ratio indicates a strong isospin and G parity violation, which is incompatible with a structure for . The decay was also observed by BaBar Collaboration babarate () at a rate:

 B(X→J/ψπ+π−π0)B(X→J/ψπ+π−)=0.8±0.3, (3)

which is consistent with the result in Eq. (2).

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 mass and the threshold: cleo (), inspired the proposal that the could be a molecular bound state with small binding energy close (); swanson (). The molecule is not an isospin eigenstate and the rate in Eq. (2) could be explained in a very natural way in this model.

Although the molecular picture is gaining attention with studies indicating that it can be a suitable description for the structure nnl (), there are also some experimental data that seem to indicate the existence of a component in its structure. In ref. Bignamini:2009sk (), a simulation for the production of a bound state with biding energy as small as 0.25 MeV, reported a production cross section that is an order of magnitude smaller than the cross section obtained from the CDF data. A similar result was obtained in ref. suzuki () in a more phenomenological analysis. However, as pointed out in ref. Artoisenet:2009wk (), a consistent analysis of the molecule production requires taking into account the effect of final state interactions of the and mesons.

Besides this debate, the recent observation, reported by BaBar babar09 (), of the decay at a rate:

 B(X→ψ(2S)γ)B(X→ψγ)=3.4±1.4, (4)

is much bigger than the molecular prediction  swan1 ():

 Γ(X→ψ(2S)γ)Γ(X→ψγ)∼4×10−3. (5)

Another interesting interpretation for the is that it could be a compact tetraquark state maiani (); tera (); tera2 (); x3872 (). In particular, Terasaki  tera2 () argues that with a tetraquark interpretation the ratio in Eq. (1) could be easily explained.

In Ref.x3872mix () the QCDSR approach was used to study the structure including the possibility of the mixing between two and four-quark states. This was implemented following the prescription suggested in oka24 () 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 suzuki (). Physically, this corresponds to a fluctuation of the 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 is the mixing of two-quarks and molecular states was also considered to investigate the radiative decay in the effective Lagrangian approach dong08 (), and to explain the data from BaBar babar09 () and Belle belle08 () using a Flatté analysis kane2 ().

In this work we will focus on the radiative decay . We use the mixed two-quark and four-quark prescription of Ref.x3872mix () to perform a QCD sum rule analysis of the radiative decay .

## Ii The mixed two-quark / four quark operator

The mixed charmonium-molecular current proposed in Ref.x3872mix () will be used to study radiative decay of the in the QCD sum rules framework.

For the charmonium part we use the conventional axial current:

 j′(2)μ(x)=¯ca(x)γμγ5ca(x). (6)

The molecule is interpolated by liuliu (); dong (); stancu ():

 j(4q)μ(x) = 1√2[(¯qa(x)γ5ca(x)¯cb(x)γμqb(x)) (7) − (¯qa(x)γμca(x)¯cb(x)γ5qb(x))],

As in Ref. oka24 () we define the normalized two-quark current as

 j(2q)μ=16√2⟨¯uu⟩j′(2)μ, (8)

and from these two currents we build the following mixed charmonium-molecular current for the :

 Jqμ(x)=sin(θ)j(4q)μ(x)+cos(θ)j(2q)μ(x). (9)

Following Ref. x3872mix () we will consider a molecular state with a small admixture of and components:

 jXμ(x)=cosαJuμ(x)+sinαJdμ(x), (10)

with , (), given by the mixed two-quark/four-quark current in Eq. (9).

## Iii The three point correlator

In this section we use QCD sum rules to study the vertex associated to the decay . The QCD sum rules approach svz (); rry (); SNB () is based on the principle of duality. It consists in the assumption that a correlation function may be described at both quark and hadron levels. At the hadronic level (the phenomenological side) the correlation function is calculated introducing hadron characteristics such as masses and coupling constants. At the quark level, the correlation function is written in terms of quark and gluon fields and a Wilson’s operator product expansion (OPE) is used to deal with the complex structure of the QCD vacuum.

The QCD sum rule calculation for the vertex is centered around the three-point function given by

 Πμνα(p,p′,q)=∫d4xd4y eip′.x eiq.yΠμνα(x,y), (11)

with

 Πμνα(x,y)=⟨0|T[jψμ(x)jγν(y)jXα†(0)]|0⟩, (12)

where and the interpolating fields are given by:

 jψμ=¯caγμca, (13)
 jγν=∑q=u,d,ceq¯qγνq, (14)

with for quarks and , and for quark ( is the modulus of the electron charge). The current is given by the mixed charmonium-molecule current in Eq. (10).

In our analysis, we consider the quarks and to be degenerate, i.e., and , then by inserting the mixed current (10) in Eq. (12), we arrive at the following relation for the correlator

 Πμνα(x,y) = esinθ3(2cosα−sinα)Πmolμνα(x,y) + e⟨¯qq⟩6√2cosθ(cosα+sinα)Πc¯cμνα(x,y).

The relation for the correlator is written in terms of the charmonium and molecule contributions. For the charmonium term we have

 Πc¯cμνα(x,y)=⟨0|T[jψμ(x)jγν(y)j′(2)α†(0)]|0⟩, (16)

and the molecular term is given by

 Πmolμνα(x,y)=⟨0|T[jψμ(x)jγν(y)j(4q)α†(0)]|0⟩, (17)

with and given by Eqs. (6) and (7) respectively.

We now proceed to the calculation of both charmonium and molecular contributions in in the OPE side. By inserting the currents of the two-quark component, , and photon, respectively defined in Eqs. (8), (13) and (14), in Eq.(16), we obtain for the charmonium contribution the following relation:

 Πc¯cμνα(x,y) = −23 Tr [γμScab(x−y)γνScbc(y)γαγ5Scca(−x) (18) + γμScac(x)γαγ5Sccb(−y)γνScba(−x+y)],

where is the full propagator of the quark (here are color indices).

For the molecular contribution we use the four-quark current defined in Eq. (7), as well as the currents for the and the photon. Inserting these currents in Eq. (17), we get

 ×Sqb′b(y)γαScba′(−x)−γμSca′c(k)γαSqab′(−y)γν× ×Sqb′b(y)γ5Scba′(k−p′)].

To evaluate the phenomenological side of the sum rule we insert, in Eq.(11), intermediate states for and . We use the following definitions:

 ⟨0|jψμ|ψ(p′)⟩=mψfψϵμ(p′); (20) ⟨X(p)|jXα|0⟩=(cosα+sinα)λqϵ∗α(p), (21)

where the meson-current coupling parameter is extracted from the two-point function, and its value was obtained in Ref. x3872mix (): . We obtain the following expression:

 Πphenμνα(p,p′,q) = −(cosα+sinα)λqmψfψϵμ(p′)ϵ∗α(p)(p2−m2X)(p′2−mψ) (22) × ⟨ψ(p′)|jγν|X(p)⟩.

The remaining matrix element can be related to the one that describes the decay :

 ⟨ψ(p′)|jγν(q)|X(p)⟩=iϵγν(q)M(X(p)→γ(q)J/ψ(p′)),

and we can define dong08 ()

 M(X(p)→γ(q)J/ψ(p′))=eεκλρσϵαX(p)ϵμψ(p′)ϵργ(q)× ×qσm2X(Agμλgακp⋅q+Bgμλpκqα+Cgακpλqμ), (24)

where are dimensionless couplings. Using this relation in Eq.(22), we can write the phenomenological side of the sum rule as:

 Πphenμνα(p,p′,q)=ie(cosα+sinα)λqmψfψm2X(p2−m2X)(p′2−mψ) (25) × (ϵαμνσqσp⋅qA+ϵμνλσp′λqσqαB−ϵανλσqμqσp′λC + ϵανλσp′λp′μqσ(C−A)p⋅qm2ψ − ϵμνλσp′λqσ(qα+p′α)(A+B)p⋅qm2X).

In the OPE side we work in leading order in and we consider condensates up to dimension five, as shown in Fig. 1. In the phenomenological side, as we can see in Eq. (25), there are five independent structures. We choose one convenient structure to determine each one of the couplings in Eq. (24). Taking the limit and doing a single Borel transform to , we arrive at a general formula for the sum rule for each structure :

 Gi(Q2)(e−m2ψ/M2−e−m2X/M2)+Hi(Q2) e−s0/M2= =¯Π(OPE)i(M2,Q2), (26)

where and gives the contribution of the pole-continuum transitions decayx (); dsdpi (); io2 (). In the following, we show the expression of the sum rules for the three structures that we have chosen to work.

### iii.0.1 Structure 1: ϵαμνσqσ

The RHS of the sum rule for the structure (structure 1) have both charmonium and molecule contributions:

 ¯ΠOPE1(M2,Q2)=−⟨¯qq⟩[sinθ(2cosα−sinα)3Q4× ×¯Π4q1(M2,Q2)+cosθ2Q2(cosα+sinα)¯Π¯cc1(M2,Q2)],

where the molecular contribution is given by

 ¯Πmol1(M2,Q2) = (1−m203Q2)∫u04m2cdu e−u/M2u× × √1−4m2cu(12+m2cu)+ + m2cm2016∫10dα1+3αα2(1−α) e−m2cα(1−α)M2.

and the charmonium contribution is

 ¯Π¯cc1(M2,Q2) = −∫s04m2cds∫u+u−due−u+sM22√λ× × (m2c+tu(t−u)λ),

where , and .

In the above expressions the parameters and are the continuum thresholds for and respectively. The limits of the integral in are:

 u±=s+t+12m2c(−st±√st(s−4m2c)(t−4m2c)). (30)

The integrals in and also obey the following conditions:

 t

Since the photon is off-shell in the vertex it is required the introduction of form factors. Then in the left hand side of the sum rule, we define the function , which is related to the form factor as:

 G1(Q2)=3√2π2(cosα+sinα)λqmψfψm2X(m2X−m2ψ)A(Q2). (32)

### iii.0.2 Structure 2: ϵμνσλp′σp′αqλ

The RHS of the sum rule for the structure (structure 2) has only molecular contribution:

 ¯ΠOPE2(M2,Q2)=m20⟨¯qq⟩Q4∫10dα1−αα e−m2cα(1−α)M2. (33)

In the left hand side of the sum rule we define the function , which is related to the sum of form factor as:

 G2(Q2)=3224√2π2(cosα+sinα)λqmψfψ(A(Q2)+B(Q2))sinθ(2cosα−sinα)m4X(m2X−m2ψ). (34)

### iii.0.3 Structure 3: ϵανλσp′λqσqμ

The RHS of the sum rule for the structure (structure 3) has only charmonium contribution:

 ¯ΠOPE3(M2,Q2)=⟨¯qq⟩∫s04m2cds∫u+u−due−u+sM22λ3/2× (35) × [tu+m2c(−s+t+u)+3 s t u(−s+t+u)λ].

The integrals in this equation obey the same relations and conditions defined for the Eq. (LABEL:2q1).

In the left hand side of the sum rule we define the function , which is related to the form factor as:

 G3(Q2)=6√2π2λqmψfψcosθm2X(m2X−m2ψ)C(Q2). (36)

## Iv Numerical analysis

The sum rules are analysed numerically using the following values for quark masses and QCD condensates x3872 (); narpdg (), and for meson masses e decay constants:

 mc(mc)=(1.23±0.05) GeV, ⟨¯qq⟩=−(0.23±0.03)3 GeV3, m20=0.8 GeV2, mψ=3.1 GeV , mX=3.87 GeV (37) fψ=0.405 GeV

The value of the angle that defines the mixing between the and has been obtained previously in Ref. decayx (); x3872mix (); maiani ():

 α=20o (38)

For the mixing angle of two and four quark states, , we use the values that were obtained in the QCD sum rules analysis of the mass of the and the decay mode x3872mix ():

 θ=(9±4)o. (39)

In the LHS of Eq. (26), the unknown functions and have to be determined by matching both sides of the sum rule. In Fig. 2, we show the points obtained if we isolate the functions in Eq. (26) and vary both and . The functions [and consequently ] should not depend on , so we limit our fit to a region where the function is clearly stable in to all values of . We can see in Fig. 2 that the regions of stability in for is , for is , and for is .

In Fig. (3) we show, through the dots, the QCDSR results for the functions as a function of . The form factors and can be easily obtained by using Eqs.(32), (34) and (36). Since the coupling constants, appearing in Eq. (24), are defined as the value of the form factors at the photon pole: , to determine the couplings , and we have to extrapolate , and 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. (3), as exponential functions:

 Gi(Q2)=g1e−g2Q2. (40)

We do the fitting for and as the results do not depend much on this parameters. The numerical values of the fitting parameters are shown in the Table 1.

From Fig. (3) we can see that the dependence of the QCDSR results for the functions are well reproduced by the chosen parametrization, in the interval , where the QCDSR are valid.

Using Eqs. (32), (34), (36) and (40) and varying in the range we get:

 A = A(Q2=0)=18.65±0.94; A+B = (A+B)(Q2=0)=−0.24±0.11; C = C(Q2=0)=−0.843±0.008. (41)

The decay width is given in terms of these couplings through dong08 ():

 Γ(X→J/ψ γ)=α3p∗5m4X((A+B)2+m2Xm2ψ(A+C)2),

where is the three-momentum of the photon in the rest frame. To compare our results with the experimental data shown in Eq. (1) we use the result for the decay width of the channel , obtained in the Ref. x3872mix (), which was computed in the same range of the mixing angle and with the same angle : . We get

 Γ(X→J/ψ γ)Γ(X→J/ψ π+π−)=0.19±0.13, (42)

which is in complete agreement with the experimental result.

## V Conclusions

We have presented a QCDSR analysis of the three-point function of the radiative decay of the meson by considering a mixed charmonium-molecular current. We find that the sum rules results in Eqs. (42) are compatible with experimental data. These results were obtained by considering the mixing angles in Eq. (10) and (9) with the values and . The present result is also compatible with previous analysis of the mass of the state and the decays into and x3872mix (), 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. x3872mix () that the is probably a state with charmonium and molecular components.

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