Diagnostic Decays of the X(3872)

The unusual properties of the X(3872) have led to speculation that it is a weakly bound state of mesons, chiefly $D^0\bar D^{0*}$. Tests of this hypothesis are investigated and it is proposed that measuring the $3\pi J/\psi$, $\gamma J/\psi$, $\gamma \psi'$, $\bar K K^*$, and $\pi\rho$ decay modes of the $X$ will serve as a definitive diagnostic of the molecule hypothesis.

This situation has led to speculation that the X could be a novel charmonium state, such as a hybrid [7], or a molecule [8]. The former scenario is viewed as unlikely because lattice computations of the lightest charmonium hybrid yield masses around 4400 MeV [9]. The molecular hypothesis is bolstered by the proximity of the X to the D 0D0 * threshold at 3871.2(7) MeV. Furthermore, the charged DD * , ρJ/ψ, and ωJ/ψ channels all lie within 7 MeV of the X mass. Thus it is natural to expect a strong admixture of these states in the X (assuming it is indeed a resonance and not a threshold enhancement). This scenario has been examined in Ref. [10] where a 6 coupled channel model with quark exchange and pion exchange interactions was diagonalised to determine the structure of the X. The conclusion was that the only viable molecular candidate is J P C = 1 ++ and is dominantly D 0D0 * with important but small admixtures of ωJ/ψ and ρJ/ψ. This state was dubbed theχ c1 . In this scenario, the decay X → π + π − J/ψ is permitted by the isospin symmetry violating ρJ/ψ component 2 of the X. However, the ωJ/ψ component can decay via 3πJ/ψ and detecting the X in this mode will be an important test of this scenario. We remark that isospin violating decay modes are a generic feature of molecular states with binding energies comparable to the mass splittings of its constituent neutral and charged mesons.
Recent experimental effort has significantly narrowed the range of options available for the structure of the X. The Belle collaboration [11] has measured the ratio of partial widths and stated that this contradicts expectations for the ψ 2 identification of the X 3 . Similarly, there is a weak signal in γχ c2 , indicating that the ψ 3 identification is unlikely. Belle also reports that the helicity angle distribution of the J/ψ in the X final state rules out J P C = 1 +− . BaBar have not seen the X in the ηJ/ψ mode [4] which is consistent with the molecular interpretation of the X. BES reports that the X is not seen in initial state radiation production [12], implying that it is inconsistent with J P C = 1 −− . Finally, CLEO has not detected the X in untagged two photon production [13], which indicates that J P C = 0 ±+ and 2 ±+ are disallowed. Together, these observations imply that the only viable nonexotic quantum numbers below J = 3 are J P C = 1 ++ . The remainder of this paper therefore concentrates on the 2 3 P 1 cc = χ ′ c1 and molecularχ c1 assignments for the X. Distinguishing these assignments is a task for experiment coupled with reliable theoretical expectations. The importance of measuring the 3πJ/ψ mode arising from the short range ωJ/ψ component of theχ c1 has already been mentioned. Here we examine the utility of a novel annihilation decay and radiative decays of the X as diagnostic tools.
Radiative decays of the molecularχ c1 to charmonium may occur via vector meson dominance in the ρJ/ψ or ωJ/ψ components of the X. Thus γJ/ψ is the only possible final state available to this mechanism. An additional mechanism has the light quarks coupling to the final state photon from the neutral and charged DD * components of theχ c1 . This diagram permits coupling to a variety of charmonia in the final state. Specific computations, presented in the next section, reveal that measurements of the rates X → γJ/ψ and X → γψ ′′ will definitively distinguish the charmonium from the molecular options for the X(3872).
The annihilation process X → KK * is studied in Section 3. This decay of the molecular state may occur via single gluon exchange but is strongly suppressed by the weakly bound nature of theχ c1 . The multigluon exchange process which contributes to cc → KK * is also suppressed but comparisons with typical charmonia indicate that charmonium rates for this reaction should be three orders of magnitude larger than that for the molecular state. A novel feature of the molecular decay is that production of charged kaons is driven by the dominant neutral DD * component of thê χ and therefore should be larger than decay into neutral kaons.

II. RADIATIVE DECAYS OF THE X
The primary mechanisms for radiative decays of theχ c1 are via vector meson dominance ( Fig. 1) and light quark annihilation (Fig. 2). The vector meson dominance (VMD) process proceeds via the ρJ/ψ and ωJ/ψ components of theχ c1 and thus contributes to Γ(X → γJ/ψ). It is clear that the amplitude must be proportional to the light vector meson wavefunction at the origin and to theχ wavefunction for the channel in question evaluated at the recoil momentum. The specific result is Here it is assumed that the ρ and ω wavefunctions are identical. The factor Z 2 α is the probability of finding the channel α in the molecular state. For a weakly bound state these probabilities are 76% (D 0D0 * ), 10% (D +D− * ), 11% (ωJ/ψ), and 0.8% (ρJ/ψ) [10] (plus 2% in D-waves). Of course relative phases of these components must be accounted for in Eq. 2. The photon momentum is denoted q. Finally, the factor of 3 appearing in this expression is a reflection of the famous VMD ratio ρ : ω : φ = 9 : 1 : 2.

FIG. 1: Vector Meson Dominance Diagram
The annihilation contribution to radiative decay illustrated in Fig. 2 proceeds via the D 0D0 * + c.c. and D −D+ * + c.c. components of theχ c1 . In evaluating this diagram I have found it convenient to assume simple harmonic wavefunctions (SHO) for the J/ψ, D, and D * mesons. These wavefunctions are specified in terms of an SHO width parameter, β (I also set β D = β D * ). This assumption permits analytic integration over eight of the nine required dimensions. The remaining integral over the radialχ wavefunction is performed numerically. The final result for the width due to annihilation is where the factors η α are a convolution of the radialχ wavefunction in the α channel with the intermediate D and D * wavefunctions and the final state charmonium wavefunction. Computations with SHO wavefunctions reveal that these factors are zero unless the final charmonium state is an S-wave. While this is not true for more realistic wavefunctions (such as Coulomb+linear) I choose not to evaluate the integrals in this case since the results will be small and will depend on model details. Relevant S-wave η factors are given by the following expressions: and These expressions depend on the quark mass ratio ρ = m c /(m c + m u ). The annihilation and vector meson dominance amplitudes are evaluated withχ wavefunctions computed in Ref. [10] and typical quark model parameters β ψ = 0.67 GeV, β D = β D * = 0.4 GeV, m c = 1.6 GeV, and m u = 0.33 GeV. The vector wavefunction at the origin is taken to be ψ ω (0) = 0.976 GeV 3/2 as determined by numerically integrating the Schrödinger equation with a Coulomb+linear+smeared hyperfine potential. This compares favourably with the SHO result of 1.17 GeV 3/2 . The final rate for X → γJ/ψ is dominated by the VMD and cross terms in the width and is presented in Table II. The rate for X → γψ ′′ and γψ 2 are zero for SHO wavefunctions since they only proceed via the annihilation diagram. Finally X → γψ ′ is also driven by the annihilation diagram and is very small.
A variety of predictions for radiative decays of the χ ′ c1 (assuming a mass of 3872 MeV) are also presented in Table  II. The fourth column summarises the results of Barnes and Godfrey [6]. These rates are computed in the impulse, nonrelativistic, zero recoil, and dipole approximations. The results for X → γJ/ψ are particularly sensitive to model details 4 . This is examined in columns five and six which present the results of two additional computations. The first, labelled [A], employs the same approximations of Barnes and Godfrey but uses meson wavefunctions computed with a simple, but accurate, Coulomb+linear+smeared hyperfine potential. It is apparent that the γJ/ψ rate is very sensitive to wavefunction details. Furthermore, one may legitimately question the use of the zero recoil and dipole approximations for the γJ/ψ mode since the photon momentum is so large in this case. The sixth column (model [B]) dispenses with these approximations, and one finds a relatively large effect for γJ/ψ. Table II makes it clear that computations of the γJ/ψ radiative transition of the χ ′ c1 are very sensitive to model details 5 . The result of Barnes and Godfrey is similar to that computed here for the molecularχ state but is much smaller than models A and B. Furthermore, the rates for γψ ′′ and γψ 2 are very small for a molecular X (at the order of eV) and quite small for a charmonium X. Perhaps the most robust diagnostic is the γψ ′ decay mode. For a molecular χ this can only proceed via the annihilation diagram of Fig. 2 and hence is very small. Clearly a measurement of the γJ/ψ and γψ ′ decay modes of the X(3872) will provide compelling clues to its internal structure.
The figures in the last column of Table I correspond to aχ state with a binding energy of 1 MeV. Changing the short range cutoff in the pion-exchange interaction of Ref. [10] modifies the binding energy and, hence, the characteristics of theχ. This affects decay rates, for example a binding energy of 4 MeV increases the γJ/ψ rate by roughly 20% and has minimal impact on the γψ ′ rate.

III. WAVEFUNCTION SUPPRESSED DECAYS: X → KK * AND πρ
Hidden flavour changing transitions of the X are interesting because they occur via cc annihilation at the origin for molecular states and via multigluon intermediate states for charmonia. One may hope that these very different mechanisms will provide additional clues to the nature of the X.
The lowest order diagram in the strong coupling in the molecular case is illustrated in Fig. 3. Here the dashed line refers to instantaneous gluon exchange. The operator giving rise to this diagram is taken to be where T a is a colour generator and K is a kernel given by br.
The constants appearing here are the strong coupling and the string tension respectively. This operator comprises a relativistic extension of the quark model and hence the computations presented here are on somewhat less firm ground than the radiative transitions of the previous section. However, note that the form used here is consistent with the Hamiltonian of QCD in Coulomb gauge.

FIG. 3: Wavefunction Suppressed Annihilation Diagram
It is likely that this diagram is much larger than the analogous transverse gluon exchange diagram because the latter contains an intermediate uūg or ddg hybrid (bound state time ordered perturbation theory is employed throughout this paper) and hence is suppressed by a large energy denominator.
The initial state of Fig. 3 is the charged or neutral DD * component of theχ; this permits the annihilation of the charmed quarks and the production of an ss pair, which materialize in the KK * final state. Notice that the production of charged kaons can only proceed via the neutral, and dominant, DD * component of theχ, while neutral kaons arise from the charged DD * component. Thus one expects Γ(X → K +K − * ) ≫ Γ(X → K 0K 0 * ) for molecular X while Γ(X → K +K − * ) ≈ Γ(X → K 0K 0 * ) for charmonium X.
The computation of Fig. 3 assumes SHO K, K * , D and D * meson wavefunctions as above. The result is where C F = (N 2 − 1)/(2N ), N is the number of colours, and η α is a factor defined by The radial wavefunction appearing in this equation is defined by φ α = Y ℓm φ αR . Evaluating Eq. 8 with typical quark model parameters b = 0.18 GeV 2 , α s = 0.5, and β K = 0.4 GeV yields The rate to neutral kaons proceeds through the smaller charged DD * component of theχ and is given by Comparable rates for P-wave charmonia are only known for the χ c0 and are Using these figures as a benchmark for 2 3 P 0 cc → KK * indicates that multigluon annihilation of charmonium χ ′ c1 is roughly 1000 times larger than the wavefunction suppressed decay of the molecularχ. Thus finding the X in KK * at the keV level may be an indication that it is a charmonium and not a molecular state.
The same computation applies to the πρ decay mode with the exception that the strange quark mass is replaced with the up or down constituent mass, the kinematics are slightly different, and both the charged and neutral DD * components of theχ wavefunction contribute to π + ρ − + c.c. and π 0 ρ 0 decays. Taking β π = β ρ = 0.4 GeV yields Γ(χ c1 → π + ρ − + c.c.) = Γ(χ c1 → π 0 ρ 0 ) = 40 eV. (13) One expects that the analogous charmonium rates are several orders of magnitude greater than this so that discovering the X in the πρ channel could be an indication that it is a charmonium state.

IV. CONCLUSIONS
Radiative decays of the X(3872) offer a promising method to distinguish the charmonium and molecular assignments for this state. Unfortunately the predicted rate for charmonium 2 3 P 1 to γJ/ψ is very sensitive to model details so that a comparison to the molecular rate is less significant that desired. Nevertheless, one expects Γ(χ c1 → γJ/ψ) < ∼ Γ(χ ′ c1 → γJ/ψ). Furthermore, determining this width for the genuine charmonium state will provide a demanding test of our understanding of heavy quark systems.
Alternatively, the charmonium width for γψ ′ is relatively stable and is much larger than the molecular rate, which can only proceed via light quark annihilation, and is therefore wavefunction suppressed. Thus collecting sufficient X events to verify that the γψ ′ final state is not seen at the 100 keV level will provide compelling evidence of the molecular nature of the X.
It has been argued that existence of 2πJ/ψ and 3πJ/ψ decay modes of the X are a strong indication of the isospin symmetry violating, and hence molecular, nature of the X. In a similar fashion, the suppressed hidden flavour changing decay X → KK * strongly favours the charged kaon final state over neutral kaons. Alternatively, one expects equal production of charged and neutral kaons from charmonium decay. An explicit computation shows that the molecular rate is very small, and probably several orders of magnitude smaller than the analogous charmonium decay. Thus a direct measurement of this rate or of the ratio of charged to neutral kaons may be a useful test of the nature of the X(3872).