QCD Sum Rules for the X(3872) as a mixed molecule-charmoniun state

# QCD Sum Rules for the X(3872) as a mixed molecule-charmoniun state

R.D. Matheus Instituto de Física, Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, SP, Brazil    F.S. Navarra Instituto de Física, Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, SP, Brazil    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 test the nature of the meson , assumed to be a mixture between charmonium and exotic molecular states with . We find that there is only a small range for the values of the mixing angle, , that can provide simultaneously good agreement with the experimental value of the mass and the decay width, and this range is . In this range we get GeV and MeV, which are compatible, within the errors, with the experimental values. We, therefore, conclude that the is approximately 97% a charmonium state with 3% admixture of 88% molecule and 12% molecule.

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

## I Introduction

Among the new hadronic states discovered in the last few years, the is one of the most interesting. It has been first observed by the Belle collaboration in the decay belle1 (). This observation was later confirmed by CDF, D0 and BaBar Xexpts (). The current world average mass is which is at the threshold for the production of the charmed meson pair . This state is extremely narrow, with a width smaller than 2.3 MeV at 90% confiedence level. Both Belle and Babar collaborations reported the radiative decay mode belleE (); babar2 (), which determines . Further studies from Belle and CDF that combine angular information and kinematic properties of the pair, strongly favor the quantum numbers or belleE (); cdf2 (); cdf3 ().

In constituent quark models bg () the masses of the possible charmonium states with quantum numbers are: and , which are much bigger than the observed mass. In view of this large mass discrepancy the attempts to understand the meson as a conventional quark-antiquark states were abandoned. The next possibility explored was to treat this state as a multiquark state, composed by , and a light quark antiquark pair. Another experimental finding in favor of this conjecture is the the fact that the decay rates of the processes and are comparable belleE ():

 X→J/ψπ+π−π0X→J/ψπ+π−=1.0±0.4±0.3. (1)

This ratio indicates a strong isospin and G parity violation, which is incompatible with a structure for .

In a multiquark approach we can avoid the isospin violation problem. The next natural question is: is the made by four quarks in a bag or by a meson-meson molecule?

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.(1) could be explained in a very natural way in this model.

Maiani and collaborators maiani () suggested that is a tetraquark. They have considered diquark-antidiquark states with and symmetric spin distribution:

 Xq=[cq]S=1[¯c¯q]S=0+[cq]S=0[¯c¯q]S=1. (2)

The isospin states with are given by:

 X(I=0)=Xu+Xd√2,X(I=1)=Xu−Xd√2. (3)

In maiani () the authors argue that the physical states are closer to mass eigenstates and are no longer isospin eigenstates. The most general states are then:

 Xl=cosθXu+sinθXd,Xh=cosθXd−sinθXu, (4)

and both can decay into and . Imposing the rate in Eq.(1), they obtain . They also argue that if dominates decays, then dominates the decays and vice-versa. Therefore, the particle in and decays would be different with maiani (); polosa () . There are indeed reports from Belle belleB0 () and Babar babarB0 () Collaborations on the observation of the decay. However, these reports (not completely consistent with each other) point to a mass difference much smaller than the predicited .

All the conclusions in ref. maiani () were obtained in the context of a quark model. Given the uncertainties inherent to hadron spectroscopy, it is interesting to confront these theoretical results with QCD sum rules (QCDSR) calculations. This was partly done in x3872 () where, using the same tetraquark structure proposed in ref. maiani (), the mass difference was computed and found to be in agreement with the BaBar measurement (). The same calculation x3872 () has obtained . In QCDSR we can also use a current with the features of the mesonic molecule of the type . With such a current the calculation reported in lnw () obtained the mass in a better agreement with the experimental mass. Therefore, from a QCDSR point of view, the seems to be better described with a molecular current than with a diquark-antidiquark current. We feel though that the subject deserves further investigation.

In this work we use again the QCDSR approach to the structure including a new possibility: the mixing between two and four-quark states. This will be implemented folowing the prescription suggested in oka24 () for the light sector. The mixing is done at the level of the currents and will be 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. As it will be seen, in order to be consistent with decay data, we must consider a second mixing between: and .

With all these ingredients we perform a calculation of the mass of the and its decay width into and .

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

There are some experimental data on the meson that seem to indicate the existence of a component in its structure. In ref. suzuki () it was shown that, because of the very loose binding of the molecule, the production rates of a pure molecule should be at least one order of magnitude smaller than what is seen experimentally. Also, 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, (5)

is much bigger than the molecular prediction  swan1 ():

 Γ(X→ψ(2S)γ)Γ(X→ψγ)∼4×10−3. (6)

While this difference could be interpreted as a strong point against the molecular model and as a point in favor of a conventional charmonium interpretation, it can also be interpreted as an indication that there is a significant mixing of the component with the molecule. Similar conclusion was also reached in refs. li1 (); li2 (). Therefore, we will follow ref. oka24 () and consider a mixed charmonium-molecular current to study the in the QCD Sum Rule framework.

For the charmonium part we use the conventional axial current:

 j′(2)μ(x)=¯ca(x)γμγ5ca(x). (7)

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

 j(4u)μ(x) = 1√2[(¯ua(x)γ5ca(x)¯cb(x)γμub(x)) (8) − (¯ua(x)γμca(x)¯cb(x)γ5ub(x))],

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

 j(2u)μ=16√2⟨¯uu⟩j′(2)μ, (9)

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

 Juμ(x)=sin(θ)j(4u)μ(x)+cos(θ)j(2u)μ(x). (10)

## Iii The two point correlator

The QCD sum rules svz (); rry (); SNB () are constructed from the two-point correlation function

 Πμν(q)=i∫d4x eiq.x⟨0|T[Juμ(x)Ju†ν(0)]|0⟩= =−Π1(q2)(gμν−qμqνq2)+Π0(q2)qμqνq2. (11)

As the axial vector current is not conserved, the two functions, and , appearing in Eq. (11) are independent and have respectively the quantum numbers of the spin 1 and 0 mesons.

The sum rules approach is based on the principle of duality. It consists in the assumption that the 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 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 phenomenological side is treated by first parametrizing the coupling of the axial vector meson , , to the current, , in Eq. (10) in terms of the meson-current coupling parameter :

 ⟨0|Juμ|X⟩=λuϵμ . (12)

Then, by inserting intermediate states for the meson , we can write the phenomenological side of Eq. (11) as

 Πphenμν(q)=(λu)2m2X−q2(−gμν+qμqνm2X)+⋯, (13)

where the Lorentz structure projects out the state. The dots denote higher mass axial-vector resonances. This ressonances will be dealt with through the introduction of a continuum threshold parameter .

In ref. 2hr () it was argued that a single pole ansatz can be problematic in the case of a multiquark state, and that the two-hadron reducible (2HR) contribution (or -wave contribution, in the present case) should also be considered in the phenomenological side. However, in ref. 2hr2 () it was shown that the 2HR contribution is very small. The reason for this is the following. The 2HR contribution, in our case, can be written as 2hr2 ():

 Π2HRμν(q) = i(λDD∗)2∫ d4p(2π)4(−gμν+pμpν/m2D∗p2−m2D∗ (14) × 1(p−q)2−m2D),

where

 ⟨0|j(4u)μ|DD∗(p)⟩=λDD∗εμ(p). (15)

Following ref. 2hr2 () the current two-meson coupling: , can be written in terms of the meson decay constant, , and the coupling of the meson with a 4-quark current. This last quantity should be very small, because the properties of the meson, both in spectroscopy and in scattering, are very well understood if it is an ordinary quark-antiquark state. Therefore, the parameter , should be very small, as in the case of the pentaquark 2hr2 (), and the 2HR contribution can be safely neglected.

In the OPE side we work up to dimension 8 at the leading order in . The light quark propagators are calculated in coordinate-space and then Fourier transformed to the momentum space. The charm quark part is calculated directly into the momentum space, with finite , and combined with the light part. The correlator in Eq. (11) can be written as:

 Πμν(q) = (⟨¯uu⟩6√2)2cos2(θ)Π(2,2)μν(q)+ (16) + ⟨¯uu⟩6√2(sin(2θ))Π(2,4)μν(q)+ + sin2(θ)Π(4,4)μν(q),

with:

 Π(i,j)μν(q)=i∫d4x eiq.x⟨0|T[j(i)μ(x)j(j)†ν(0)]|0⟩. (17)

After making a Borel transform of both sides, and transferring the continuum contribution to the OPE side, the sum rule for the axial vector meson up to dimension-eight condensates can be written as:

 (λu)2e−m2X/M2= =(⟨¯uu⟩6√2)2cos2(θ)Π(2,2)1(M2)+ +sin2(θ)Π(4,4)1(M2), (18)

where:

 Π(2,2)1(M2)=∫s04m2cds e−s/M2ρ(22)pert(s)+Π(22)⟨G2⟩(M2), (19)
 Π(2,4)1(M2)=∫s04m2cds e−s/M2ρ(24)⟨¯uu⟩(s)+Π(24)⟨¯uGu⟩(M2), (20)
 Π(4,4)1(M2) = ∫s04m2cds e−s/M2[ρ(44)pert(s)+ρ(44)⟨¯uu⟩(s)+ (21) + ρ(44)⟨¯uu⟩2(s)+ρ(44)⟨G2⟩(s)+ρ(44)⟨¯uGu⟩(s)]+ + Π(44)⟨¯uu⟩⟨¯uGu⟩(M2),

and

 ρ(22)pert(s)=s4π2(1−4m2cs)32, (22)
 Π(22)⟨G2⟩(M2) =−⟨g2G2⟩3⋅25π2∫10dα[2α(1−α)M2+m2cα(1−α)M2+ +2m2c(2α−1)m2c+α(α2−1)M2M4α3(α−1)]e−m2cα(1−α)M2,
 ρ(24)⟨¯uu⟩(s)=−⟨¯uu⟩6√2ρ(22)pert(s), (24)
 Π(24)⟨G2⟩(M2)=−⟨¯uu⟩6√2Π(22)⟨G2⟩(M2) (25)
 Π(24)⟨¯uGu⟩(M2) =5⟨¯qgsσ⋅Gq⟩3⋅26√2π2∫10dαm2c1−αe−m2cα(1−α)M2, (26)
 ρ(44)pert(s) =3212π6∫αmaxαmindα∫βmaxβmindβ1−(α+β)2α3β3K4(α,β),
 ρ(44)⟨¯uu⟩(s) =−3mc⟨¯uu⟩27π4∫αmaxαmindα∫βmaxβmindβ(1+α+β)αβ2K2(α,β),
 ρ(44)⟨¯uu⟩2(s)=m2c24π2⟨¯uu⟩2√1−4m2cs, (29)
 ρ(44)⟨G2⟩(s) =⟨g2G2⟩211π6∫αmaxαmindα∫βmaxβmindβ[m2c(1−(α+β)2)α3 −1−2α−2βαβ2K(α,β)]K(α,β),
 ρ(44)⟨¯uGu⟩(s) =−3mc28π4⟨¯ugσ⋅Gu⟩∫αmaxαmindα[2(m2c−α(1−α)s)1−α (31) −∫βmaxβmindβ(1−2α+ββ)K(α,β)β],
 Π(44)⟨¯uu⟩⟨¯uGu⟩(M2) =−m2c⟨¯uu⟩⟨¯ugsσ⋅Gu⟩25π2∫10dα[α(1−α)M2+m2cα(1−α)M2 (32) −11−α]e−m2cα(1−α)M2.

The integration limits are:

 αmin=1−√1−4m2cs2,αmax=1+√1−4m2cs2
 βmin=ααq2m2c−1,βmax=1−α

and we define .

By taking the derivative of Eq. (18) with respect to and dividing the result by Eq. (18) we can obtain the mass of without worrying about the value of the meson-current coupling . The expression thus obtained is analised numerically using the following values for quark masses and QCD condensates x3872 (); narpdg ():

 mc(mc)=(1.23±0.05) GeV, ⟨¯uu⟩−(0.23±0.03)3 GeV3, ⟨¯ugσ.Gu⟩=m20⟨¯uu⟩, m20=0.8 GeV2, ⟨g2G2⟩=0.88  GeV4. (33)

In Fig. 1 we show the contributions of the terms in Eqs. (22) to (32) grouped by condensate dimensions divided by the RHS of Eq. (18). We have used GeV and , but the situation does not change much for other choices of these parameters. It is clear that the OPE is converging for values of GeV and we will limit our analysis to that region.

The upper limit to the value of comes by imposing that the QCD pole contribution should be bigger than the continuun contribution. The maximum value of that satisfies this condition depends on the value of , being more restrictive for smaller . In Fig. 2 we show a comparison between the pole and continuun contributions for the smaller we will be considering () and . The condition obtained from Fig. 2 is GeV, but in this case, the dependence on the choice of is very strong. Taking into account the variation of we have determined that, for , the QCDSR are valid in the following region:

 2.6 GeV2≤M2≤3.0 GeV2 (34)

In Fig. 3, we show the meson mass in this region. We see that the results are reasonably stable as a function of .

From Fig. 3 we obtain where the error includes the variation of both and . If we also take into account the variation of in the region we get:

 mX=(3.77±0.18) GeV, (35)

which is in a good agreement with the experimental value. The value obtained for the mass grows with the value of the mixing angle , but for it reaches a stable value being completely determined by the molecular part of the current.

From Eq. (18) we can also obtain by fixing equal to the experimental value (). Using the same region in , and that we have used in the mass analysis we obtain:

 λu=(3.6±0.9).10−3 GeV5. (36)

## Iv Decay of the X(3872) and the three point correlator

As discussed in Sec. I, one of the most intriguing facts about the meson is the observation, reported by the BELLE collaboration belleE (), that the decays into , with a strength that is compatible to that of the mode, as given by Eq .(1). This decay suggests an appreciable transition rate to and establishes strong isospin violating effects. It still does not completely exclude a interpretation for since the origin of the isospin and G parity non-conservation in Eq. (1) could be of dynamical origin due to mixing tera (). However, the observation of the ratio in Eq. (1) is an important point in favor of the molecular picture proposed by Swanson swan1 (). In this molecular picture the is mainly a molecule with a small but important admixture of and components.

It is important to notice that, although a molecule is not an isospin eingenstate, the ratio in Eq. (1) can not be reproduced by a pure molecule. This can be seen through the observation that the decay width for the decay where for is given by maiani (); decayx ()

 dΓds(X→J/ψf)=18πm2X|M|2BV→F (37) × ΓVmVπp(s)(s−m2V)2+(mVΓV)2,

where

 p(s)=√λ(m2X,m2ψ,s)2mX, (38)

with . The invariant amplitude squared is given by:

 |M|2=g2XψVf(mX,mψ,s), (39)

where is the coupling constant in the vertex and

 f(mX,mψ,s)=13(4m2X−m2ψ+s2+(m2X−m2ψ)22s (40) + (m2X−s)22m2ψ)m2X−m2ψ+s2m2X.

Therefore, the ratio in Eq. (1) is given by:

 Γ(X→J/ψπ+π−π0)Γ(X→J/ψπ+π−)=g2XψωmωΓωBω→πππIωg2XψρmρΓρBρ→ππIρ, (41)

where

 IV = ∫(mX−mψ)2(nmπ)2ds(f(mX,mψ,s) (42) × p(s)(s−m2V)2+(mVΓV)2).

Using , , , , and we get

 Γ(X→J/ψπ+π−π0)Γ(X→J/ψπ+π−)=0.118(gXψωgXψρ)2. (43)

The couplings, , can be evaluated through a QCDSR calculation for the vertex, , that centers in the three-point function given by

 Πμνα(p,p′,q)=∫d4xd4y eip′.x eiq.yΠμνα(x,y), (44)

with

 Πμνα(x,y)=⟨0|T[jψμ(x)jVν(y)jXα†(0)]|0⟩, (45)

where and the interpolating fields are given by:

 jψμ=¯caγμca, (46)
 jVν=NV2(¯uaγνua+(−1)IV¯daγνda), (47)

with , , and . If is a pure molecule, is given by Eq. (8). In this case the only difference in the OPE side of the sum rule is the factor and, therefore, regardless the approximations made in the OPE side and the number of terms considered in the sum rule one has

 ΠVμνα(p,p′,q)=NVΠOPEμνα(p,p′,q). (48)

To evaluate the phenomenological side of the sum rule we insert, in Eq.(45), intermediate states for , and . We get decayx ():

 Π(phen)μνα(p,p′,q)=iλXmψfψmVfV gXψV(p2−m2X)(p′2−m2ψ)(q2−m2V) (49) × (−ϵαμνσ(p′σ+qσ)−ϵαμσγp′σqγqνm2V − ϵανσγp′σqγp′μm2ψ).

Therefore, for a given structure the sum rule is given by:

 iλXmψfψmVfV gXψV(p2−m2X)(p′2−m2ψ)(q2−m2V)=NVΠOPE(p,p′,q), (50)

from where, considering one gets:

 gXψωfωgXψρfρ=NωNρ=13. (51)

Using and we obtain

 gXψωgXψρ=1.14, (52)

and using this result in Eq. (43) we finally get

 Γ(X→J/ψπ+π−π0)Γ(X→J/ψπ+π−)≃0.15. (53)

It is very important to notice that this is a very general result that does not depend on any approximation in the QCDSR. This result shows that the admixture of and components in the molecular model of ref.swan1 () is indeed very important to reproduce the data in Eq. (1). It is also important to notice that, in a QCDSR calculation of the decay rate , the admixture in the molecule, as given by Eq. (10), does not solve the problem of geting the ratio in Eq.(1). This can be seen by using, in Eq. (45), , with given by Eq. (10). One gets:

 Πμνα(x,y) = ⟨¯uu⟩2√6cos(θ)Πc¯cμνα(x,y) (54) + sin(θ)Πmolμνα(x,y),

where

 Πc¯cμνα(x,y)=⟨0|T[jψμ(x)jVν(y)j′(2)α†(0)]|0⟩, (55)

and

 Πmolμνα(x,y)=⟨0|T[jψμ(x)jVν(y)j(4u)α†(0)]|0⟩, (56)

with and given by Eqs. (7) and (8). Using the currents in Eqs.(47) and (46) for the mesons and , it is easy to see that

 Πc¯cμνα(x,y) = NV2 Tr [γμScac(x)γαγ5Scca(−x)]× (57) × Tr [γνSubb(0)+(−1)IVγνSdbb(0)].

For with the result in Eq. (57) is obviously zero due to isospin conservation, in the case that the quark and are degenerate. However, even for (), the result in Eq. (57) is zero because . Therefore, in the OPE side, the three-point function is given only by the molecular part of the current in Eq (10):

 Πμνα(x,y)=sin(θ)Πmolμνα(x,y), (58)

that can not reproduce the experimental observation in Eq. (1), as demonstrated above.

In the following, to be able to reproduce the data in Eq.(1), instead of the admixture of and components to the molecule, as done by Swanson swan1 (), we will consider a small admixture of and components. In this case, instead of Eq.(10) we have

 jXμ(x)=cosαJuμ(x)+sinαJdμ(x), (59)

with and given by Eq.(10).

If we consider the quarks and to be degenerate, i.e., and , the change in Eq.(10) to Eq.(59) does not make any difference in the results in Sec. III.

By inserting , given by Eq. (59), in Eq. (45) and considering the quarks and to be degenerate, one has

 Πμνα(p,p′,q) = sin(θ)NV2√2(cosα (60) + (−1)IVsinα)ΠOPEμνα(p,p′,q),

with

 ΠOPEμνα(p,p′,q)=∫ d4u∫ d4k(2π)4( Tr [γμSca′c(k)γ5× (61) × Sqab′(−y)γν