# Hadronic Decays of the to in Effective Field Theory

###### Abstract

The decays of the to -wave quarkonia are calculated under the assumption that it is a shallow bound state of neutral charmed mesons. The is described using an effective theory of nonrelativistic mesons and pions (X-EFT). We calculate decays by first matching heavy hadron chiral perturbation theory (HHPT) amplitudes for onto local operators in X-EFT, and then using these operators to calculate the decays. This procedure reproduces the factorization theorems for decays to conventional quarkonia previously derived using the operator product expansion. For single pion decays, we find nontrivial dependence on the pion energy from HHPT diagrams with virtual mesons. This nontrivial energy dependence can potentially modify heavy quark symmetry predictions for the relative sizes of decay rates. At leading order, decays to final states with two pions are dominated by the final state , with a branching fraction just below that for the decay to . Decays to all other final states with two pions are highly suppressed.

The Choi:2003ue ; Acosta:2003zx ; Abazov:2004kp is a novel charmonium state that lies very close to the threshold. The closeness of the to this threshold has prompted numerous authors to suggest that it is a molecular bound state of , for reviews see Refs. Voloshin:2006wf ; Godfrey:2008nc . The strongest evidence for the molecular hypothesis is the large ratio of the branching fractions for the decays to plus two or three pions Abe:2005ix

(1) |

The final states have opposite G-parity which implies that the does not have definite isospin. Another possibility, suggested in Ref. Gamermann:2007fi , is that there are actually two nearly degenerate states with opposite G-parity. Measurements of the invariant mass distributions of the pions indicate that these decays proceed through for the final state, and through for the final state, suggesting that the couples to and channels with roughly equal strength. This is not possible for a conventional state. The observed branching ratio can be understood if the is comprised primarily of neutral mesons, which is expected since the charged - threshold is far () above the mass.

If the is a molecular state then the large distance behavior of the wavefunction is determined entirely by the binding energy and effective range theory (ERT) can be used to calculate some properties of the . ERT exploits the fact that at long distances the two-body wavefunction of the must be

(2) |

where is the separation of the meson constituents and , where is the binding energy of the , and is - reduced mass. Parameter-free predictions for the decays Voloshin:2003nt and Voloshin:2005rt have been obtained using ERT. Experimental measurement of these decays consistent with these calculations would confirm the molecular hypothesis. Unfortunately, data which can test these calculations is not currently available. For example, the photon energy spectrum in the decay near the kinematic endpoint can be computed in terms of the binding energy, but this spectrum has not been measured yet. The ratio of branching ratios Gokhroo:2006bt

(3) |

has been measured but cannot be calculated within ERT since the decay is sensitive to the short-distance structure of the .

Many of the observed decays of the are to final states that include conventional quarkonium such
as , ,
and . These decays necessarily involve shorter distance
scales where the description of the as a loosely bound state of and
mesons is no longer valid. To make quantitative calculations for these decays one can use factorization
theorems that separate long-distance physics () from physics at shorter
distance scales () Braaten:2005jj ; Braaten:2006sy .
Here can be determined from the binding energy using the known masses
of the , , and . For and , we use
and , respectively Amsler:2008zz .
For the we use the mass obtained in measurements of decays
to final states only, which yields
Yao:2006px . ^{3}^{3}3Interpretation of mass measurements in decays to final states
is complicated by the possibility of two distinct states Gamermann:2007fi ; Amsler:2008zz as well as threshold
enhancements due to the large scattering length in channel. We thank E. Braaten for discussion on this last point.
Using these numbers we find , so .
The factorization theorems
express the rate for (where is the final state) in terms of the the rate
for times universal factors that depend on the binding energy of the .
These factorization theorems have been used to analyze decays into plus light hadrons
in Ref. Braaten:2005ai .

Recently, Ref. Dubynskiy:2007tj has studied the decays and argued that measurement of the partial rates for these decays can discriminate between conventional charmonium and molecular interpretations of the . Decays to are particularly interesting since the are in a heavy quark multiplet and therefore heavy quark symmetry can be used to make predictions for the relative sizes of the partial rates.

In this paper, we study the strong decays using a combination of heavy hadron chiral
perturbation theory (HHPT) Wise:1992hn ; Burdman:1992gh ; Yan:1992gz and X-EFT, an effective field theory for the
developed in Ref. Fleming:2007rp . HHPT is an effective theory for low energy hadronic interactions of
mesons containing heavy quarks that incorporates the constraints of the heavy quark and chiral symmetries of QCD. We use
HHPT to calculate the ^{4}^{4}4Here and below decays related by charge
conjugation are implied. transition amplitudes.
X-EFT Fleming:2007rp generalizes the effective theory developed for shallow nuclear
bound states in Ref. Kaplan:1998tg ; Kaplan:1998we ; vanKolck:1998bw . In this theory, modes have momenta of order
so the degrees of freedom are non-relativistic charmed mesons and non-relativistic pions. The decays
proceed through local operators that couple the and to the
and one or two pions. We use the HHPT calculation of the
amplitudes to fix the coefficients of these operators in the X-EFT Lagrangian, then calculate the decays of the .
The resulting decay rates are consistent with the factorization theorems for
decays proven in Refs. Braaten:2005jj ; Braaten:2006sy .

In the leading order (LO) diagrams for , virtual charmed meson propagators introduce dependence on the pion momentum which modifies the usual dependence expected for a -wave single pion decay. This can give significant corrections to the relative rates predicted in Ref. Dubynskiy:2007tj . We also calculate the differential decay rates for . At LO in HHPT, amplitudes for and are given in terms of a single unknown coupling in the HHPT lagrangian. Therefore, at LO, the normalization of the decay rates can be given in terms of the rates for . We find that is the dominant decay mode with two pions in the final state, with a branching fraction that is just times the branching fraction for . All other decays to final states with two pions are considerably smaller. In some of the tree level diagrams for , a virtual meson can go nearly on-shell, with the potential divergence cutoff by the width of the . The resulting contribution for the decay rate is enhanced by , where is the typical energy of the pion in the final state (about 200 MeV) and , the width of the , is estimated to be about 68 keV Hu:2005gf . This enhancement only occurs for the final state, total partial decay rates to other final states with two pions are smaller by orders of magnitude.

The LO HHPT lagrangian for mesons containing heavy quarks or antiquarks at rest is

(4) | |||||

We use the two component notation of Ref. Hu:2005gf . The field is given by

(5) |

where annihilates mesons and annihilates mesons. The subscript is an index, and for neutral mesons. The field for antimesons is

(6) |

The field is the axial current of chiral perturbation theory, , where is the pion decay constant and are the Goldstone boson fields. The axial coupling, , is taken to be and is the hyperfine splitting.

To calculate the amplitude for we need to include the in the HHPT lagrangian. The are degenerate in the heavy quark limit and form a heavy quark multiplet. The fields for have been constructed in a covariant formalism in Ref. Casalbuoni:1992dx . In our two component notation, the fields are represented by

(7) | |||||

The transformation properties of the heavy quark and quarkonium fields under the various symmetries are:

Note that under charge conjugation, the components of transform as and . Therefore, the even combination of mesons that couples to the is . In the first line, () are () rotation matrices related by . In the second line, is a rotation matrix acting only on the heavy-quark spin, is a rotation matrix acting only on the heavy-antiquark spin and in the last line is an matrix.

The lagrangian coupling the to heavy mesons is

(8) |

The leading contribution to the amplitude for comes from the tree level diagrams in Figs. 1a), b), and c). In these diagrams, the coupling of to mesons comes from the first operator in Eq. (8). Fig. 1a) contributes to decays, Fig. 1b) contributes to both and decays, and Fig. 1c) contributes to decays only. In the HHPT power counting, one takes , and expands amplitudes in powers of . The graphs in Figs. 1a)-c) are since the coupling of the mesons to has no derivatives and is therefore , the axial coupling of the mesons is , and the heavy meson propagator is . Fig. 1d) shows the contribution to the amplitude from the second operator in Eq. (8). This contribution is because this operator contains one derivative. Chiral loop corrections to the LO diagrams in Eq. (1) are suppressed by and therefore subleading to the diagram in Fig. 1d). It is also clear that higher dimension operators that couple the mesons to either contain two spatial derivatives or an insertion of the light quark mass matrix and are suppressed relative to the leading operator in Eq. (8) by at least . Therefore the diagrams in Fig. 1 give the complete HHPT calculation to next-to-leading (NLO) order.

We obtain

(9) |

The results for the charge conjugate initial states are identical. These amplitudes must be multiplied by a factor of to account for the non-relativistic normalization of the HHPT fields.

The virtual charmed mesons in Figs. 1a)-c) are off-shell by , where varies from 432 MeV to 305 MeV in the three decays. X-EFT contains non-relativistic , , and , and total energies of these particles are assumed to be within a few MeV of the threshold. Thus we must match the tree level diagrams in Fig. 1 onto local operators in X-EFT. We will illustrate this for the decay to , the generalization to other decays is straightforward. The operator in X-EFT which reproduces the amplitudes for and is

(10) |

where we have dropped the labels on the meson fields that are implicitly neutral. Note, when evaluating the amplitude we must multiply the naive Feynman rule obtained from Eq. (10) by is included to compensate for the differing normalizations in the effective field theory and HHPT. Comparing with Eq. (Hadronic Decays of the to in Effective Field Theory) (times ), we find for , to account for the normalizations of the fields in the effective theory, which are all nonrelativistic, including the pion. The factor of

(11) |

Now we consider the decay mediated by this operator. The one-loop diagram contributing to this decay is depicted in Fig. 2. The decay matrix element is obtained from the diagram in Fig. 2 after dividing by the wavefunction renormalization factor Fleming:2007rp . After summing over both channels, the result is

(12) |

In evaluating the diagram in Fig. 2 we have used

(13) | |||||

It is straightforward to evaluate the rates:

(14) |

where , , , and the functions are given by

(15) |

Using these results we can calculate the partial rates for . At LO () we find (denoting )

(16) |

The energy dependence from the virtual pion propagators makes a significant change in the predictions for the relative sizes of the partial rates. The NLO predictions depend on the ratio which is undetermined. The ratio has dimensions of inverse mass, when is varied from to , we find:

(17) |

For the largest values of , we obtain predictions for the relative sizes of the partial rates similar to those found in Ref. Dubynskiy:2007tj . As decreases, the predictions tend to those appearing in Eq. (16). Experimental measurement of the relative sizes of the partial rates for decays to can be used to determine .

Refs. Braaten:2005jj ; Braaten:2006sy derived factorization theorems for decaying into quarkonium states and light hadrons. Note that the first term in the decay rates in Eq. (14) can be written as

(18) |

where is a polarization vector. Thus we see that the expressions for the partial widths factor into the product of long-distance matrix elements, given in Eq. (18), times short-distance coefficients, which are proportional to the cross sections for . This is the content of the factorization theorems of Refs. Braaten:2005jj ; Braaten:2006sy . A main point of this paper is that HHPT can be used to calculate the short-distance coefficients in these factorization theorems.

Next we turn to the calculation of the three-body decays, . In Fig. 3 we show the diagrams contributing to the transition amplitudes for . These amplitudes are matched onto local operators in X-EFT, then these local operators are used to calculate the decays , in a manner identical to the procedure used to calculate . Here we give our results for the differential partial decay rates.

The differential decay rate for is given by

(19) |

where

(20) | |||||

Here, refers to the energy (three-momentum) of one of the and . The partial decay rates are symmetric under . In these decays is equal to the neutral hyperfine splitting, .

The rates for can also be calculated. The differential rates for are bigger by a factor of two than the corresponding rates for due to isospin factors in the diagram. Furthermore, the factor of can be , or , depending on the diagram, so the expressions above (times 2) can only be used if one neglects isospin breaking in the heavy meson hyperfine splittings.

The differential decay rate for obtained from diagrams in Figs. 3b),c),e), and f) is

where

(22) |

In Figs. 3c) and 3f), the virtual meson can go on-shell leading to the poles of
in the expression for the decay rate. In the case of final states, the virtual charged meson cannot go on-shell
but can be as small as 2 MeV. The pole in the decay amplitude is regulated by putting the
external on its complex energy mass shell, . ^{5}^{5}5We thank
Eric Braaten for suggesting this. The width of the has not been measured. If one uses HHPT
to relate the width of the to the measured width of the and the measured strong decay branching
fractions for both and , one finds Hu:2005gf . Since
this scale is orders of magnitude smaller than all other scales in the problem, we will keep only terms
which do not vanish in the limit . For terms with single poles in ,
this amounts to using a principal value prescription. When integrating over the double poles,
we replace in the denominators of Eq. (Hadronic Decays of the to in Effective Field Theory),
then perform the energy integrals using the formula

(23) | |||||

where the elipsis denotes terms that can be neglected when . Keeping only the first term in Eq. (23) is called the narrow width approximation which turns out to be an excellent approximation for .

One might also worry about contributions from interference terms that contain single poles in and . These are of the form

The first term on the righthand side clearly yields two single poles which should be dealt with via the principal value prescription in the limit . The second term only has a nonvanishing contribution (in the limit ) with support at , which lies outside available phase space.

Finally, the differential decay rate for is

where

(26) |

The normalization of the three-body decay rates is unknown but can be given in terms of the LO chiral perturbation theory expressions for the two-body decay rates computed earlier. Integrating over three-body phase space we find

(27) |

Here we have used and . The subscript LO emphasizes that this result only holds at LO in PT. At NLO the prediction depends on the unknown parameter, .

We have not numerically computed the corresponding branching fraction ratios for . For final states with or , isospin violation is a small effect so we expect