Decays of Pentaquarks in Hadrocharmonium and Molecular Pictures

Decays of Pentaquarks in Hadrocharmonium and Molecular Pictures

Michael I. Eides meides@g.uky.edu Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Petersburg Nuclear Physics Institute, Gatchina, 188300, St.Petersburg, Russia    Victor Yu. Petrov Victor.Petrov@thd.pnpi.spb.ru Petersburg Nuclear Physics Institute, Gatchina, 188300, St.Petersburg, Russia
Abstract

We consider decays of the hidden charm LHCb pentaquarks in the hadrocharmonium and molecular scenarios. In both pictures the LHCb pentaquarks are essentially nonrelativistic bound states. We develop a semirelativistic framework for calculation of the partial decay widths that allows the final particles to be relativistic. Using this approach we calculate the decay widths in the hadrocharmonium and molecular pictures. Molecular hidden charm pentaquarks are constructed as loosely bound states of charmed and anticharmed hadrons. Calculations show that molecular pentaquarks decay predominantly into states with open charm. Strong suppression of the molecular pentaquark decays into states with hidden charm is qualitatively explained by a relatively large size of the molecular pentaquark. The decay pattern of hadrocharmonium pentaquarks that are interpreted as loosely bound states of excited charmonium and nucleons is quite different. This time dominate decays into states with hidden charm, but suppression of the decays with charm exchange is weaker than in the respective molecular case. The weaker suppression is explained by a larger binding energy and respectively smaller size of the hadrocharmonium pentaquarks. These results combined with the experimental data on partial decay widths could allow to figure out which of the two theoretical scenarios for pentaquarks (if either) is chosen by nature.

I Introduction

Pentaquarks discovered by the LHCb collaboration LHCb2015 (); LHCb2016 () are the first experimental sighting of exotic baryons. It is probably not by chance that these baryons contain a heavy quark-antiquark pair, with quark masses larger than the scale of strong interactions. Internal structure of the LHCb pentaquarks remains at this moment unknown. Numerous models of the exotic pentaquarks were proposed in the literature, see, e.g., recent reviews rflremess2017 (); als2017 (); aeapadp2017 (); slotsdz2018 (); fkgchugm2018 (); mkjlrts2018 () and references therein.

We will concentrate on the popular molecular and hadrocharmonium scenarios for the LHCb pentaquarks as they were realized in epp2016 (); epp2018 () (see also abfs2018 ()). Neither of these scenarios can be justified on purely theoretical grounds, both are based on some physically reasonable conjectures about the nature of QCD at low energies. Both in the hadrocharmonium and the molecular pictures pentaquark is assumed to be a nonrelativistic bound state of two hadrons. The main difference between the two models is in the nature of forces that bind constituents into a pentaquark. The idea of the hadrocharmonium picture sibvol2005 (); dubvol2008 (); livol2014 () is that almost static heavy quark and antiquark inside an exotic baryon form a small color singlet state – one of excitations of charmonium. Light valence quarks inside hadrocharmonium also form a color singlet state (nucleon) and occupy a much larger volume. Interaction between an almost static color singlet heavy quark-antiquark pair and a large color singlet nucleon is due to the long range color dipole forces and effectively the small static pair probes the long wavelength gluon field inside the large light nucleon. Heavy quarkonium interaction with nuclei was considered in sjbisgt (); lms1992 (), see also references in volsh2008 (). A QCD motivated potential that depends on the charmonium chromoelectric polarizability and nucleon stress-energy distribution describes charmonium-nucleon interaction, and one can find the spectrum of hidden charm baryons solving the Schrödinger equation epp2016 (); epp2018 (). Literally, the hadrocharmonium picture is justified in the large and heavy quark limit when the mass of the nucleon becomes large and its size remains constant, while the heavy quark-antiquark pair occupies a small volume and is effectively static sibvol2005 (); dubvol2008 ().

The molecular scenario of hidden charm pentaquarks initiated in volok1976 () is qualitatively vastly different. In this scenario heavy quark and valence light quark(s) form a color singlet open charm heavy hadron, while the heavy antiquark forms another open charm hadron with the remaining light valence quark(s). These open charm hadrons interact via exchange of light mesons and form a loosely bound pentaquark where the open charm constituent hadrons and, respectively, heavy quark and antiquark are at rather large distances. The problem with this scenario is that meson exchanges generate attraction at large distances but are too singular at short distances and fail to hold the constituents far enough to avoid fall to the center. Some kind of hard core should arise and meson exchanges do not provide any effective repulsion at small distances. Therefore the hard core is not under theoretical control while the wave function in the molecular scenario tends to be concentrated there and critically depends on the hard core properties, see, e.g., epp2018 () and references in the reviews als2017 (); aeapadp2017 (); fkgchugm2018 ().

Currently both the molecular and hadrocharmonium descriptions of the LHCb pentaquarks are plausible, one cannot choose between them on purely theoretical grounds. Taking into account uncertainty of the theoretical situation, one needs to find experimentally observable signatures that could help to figure out which of the two scenarios (if any) is realized by nature. In principle, there are many ways to explore internal structure of hadrons, the most straightforward approach is just to measure their form factors. Information on the electromagnetic form factors of pentaquarks could immediately resolve the confrontation of the hadrocharmonium and molecular scenarios. However, one cannot expect any experimental data on the form factors of the LHCb pentaquarks any time soon. The next best option to explore internal structure of pentaquarks is to measure decays widths.

We expect that the dominant contributions to the total width come from two-particle decays. In the hadrocharmonium picture decays with emission of additional pions are strongly suppressed due to small phase volume and pseudogoldstone nature of pions epp2016 (). The constituents of the molecular pentaquark are unstable with respect to decays and , and have finite but small widths. Three-particle decays are banned kinematically, MeV. Decays are allowed kinematically, MeV but they are suppressed due to a small available phase volume and derivative coupling of pions.

Both in the hadrocharmonium and molecular pictures there are two qualitatively different classes of two-particle pentaquark decay processes. Decays of one kind occur without charm exchange between the constituents and the decay products carry the same charm as the constituents. In decays of the other kind charm is exchanged and the decay products have charm quantum numbers that do not coincide with the ones of the constituents.

Calculations of the pentaquark decays are impeded by numerous obstacles: apparent ultraviolet divergences, uncertainty of the cutoff momenta, need to introduce more or less arbitrary form factors, etc. We describe decay processes of nonrelativistic loosely bound pentaquarks by -channel exchanges between the constituent hadrons111Processes with the -channel annihilation of heavy -quarks are suppressed due to the Zweig-Okubo-Iizuka rule.. In transitions without charm exchange interaction is due to the lightest mesons without open charm. In the case when charm of the constituents changes they exchange by the lightest mesons with open charm. A naive expectation is that in each case (hadrocharmonium and molecular pentaquarks) decays without charm exchange dominate and decays with charm exchange are suppressed. This pattern of decays could allow to choose between the hadrocharmonium and molecular pictures of pentaquarks if and when the experimental data for decays will be available.

Let us quantify these expectations. Notice that to exchange charm the constituents should come very close to each other, at a relative distance . The probability of this to happen in a nonrelativistic bound state is proportional to , where is the bound state wave function. But , where , is the reduced mass of the system and is the binding energy. Then suppression of decays with exchange of charm is described by the factor

(1)

In a hadrocharmonium pentaquark is about the nucleon mass and in a molecular pentaquark . For the constructed in epp2016 (); epp2018 () binding energy is MeV in the hadrocharmonium case, and it is MeV in the molecular case. At face value suppression of decays with charm exchange is expected in both pictures and it is stronger in the molecular picture. We will see below that these expectations hold and discuss what happens.

Our principal goal is to find out if measurements of partial widths for decays in the channels with open and hiddent charm can help to figure our which of the two scenarios (hadrocharmonium and molecular) of the hidden charm pentaquarks is realized in nature. To this end we develop a semirelativistic approach to calculation of the decays. Let us emphasize that despite bound states both in the hadrocharmonium and the molecular pictures are nonrelativistic, loop momenta are in principle arbitrary and the final decay momentum is sometimes relativistic. In the semirelativistic approach we make a physically reasonable assumption that the intermediate virtual particles in the loop diagrams are always not far from their mass shell what allows to threat them nonrelativistically. On the other hand, our approach allows to treat the exchanged particle as well as the final particles relativistically. Below we consider decays of the hadrocharmonium and molecular pentaquarks from epp2018 () in this approach. We start with the basic features of the semirelativistic approximation that allows one to calculate the pentaquark decays with a reasonable accuracy. We use Feynman diagrams to derive the interaction potentials for different decays, calculate decay widths of hadrocharmonium and molecular pentaquarks222Decays of pentaquarks in the molecular picture were discussed in the literature earlier, see, e.g., lsgz2017 (); ludong2016 (); sgxz2016 (); sl2018 (); kayshs2018 () and references therein. To the best of our knowledge decays in the hadrocharmonium picture were never discussed before., make predictions for relative rates of different decays in each picture and compare the patterns of decays in hadrocharmonium and molecular scenario.

Ii Semirelativistic Approximation for Pentaquarks Decays

ii.1 Kinematics

The first task is to derive a practical general formula for calculation of the pentaquark decays. We consider pentaquarks as loosely bound states of two particles with binding energy () much smaller than the reduced mass of the constituents, . The constituent particles are close to the mass shell and are nonrelativistic, . In the case of the LHCb pentaquark constructed as a bound state of and the nucleon epp2016 (); epp2018 () MeV, MeV, and the relativistic correction to the binding energy is about %. The accuracy of the nonrelativistic approximation for other systems and processes considered below is roughly the same. We will use the nonrelativistic approximation in calculation of widths of loosely bound states ignoring off-masshellness of the constituents. We expect the obtained results to have error bars about 6-8 %.

Pentaquark decays both in the hadrocharmonium and molecular pictures are due to the diagrams with the -channel exchange of the type represented in Fig. 1, where and are the pentaquark constituents, and and are the decay products. To make the discussion more transparent we temporarily ignore spins of all particles. The final particles with masses and as well as the exchanged virtual particle , could have masses significantly smaller than the masses of the constituents and are not necessarily nonrelativistic. We need to use relativistic kinematics for these particles. Then the decay width of the pentaquark has the form

(2)

where is the three-momentum of the final particle and we integrate over its directions, is the normalized nonrelativistic wave function of the initial pentaquark (a loosely bound state of particles and ) in its rest frame, and the effective potential ( are the respective coupling constants) is in the general case a function of the relative coordinate and the final momentum . Notice the relativistic energies in Eq. (2) instead of the masses in the standard nonrelativistic formula. They arise because the final particles could be relatively light and relativistic.

Figure 1:

The integral in Eq. (2) can be simplified when the bound state wave function is a superposition of terms with different angular momenta and is a central potential. In such case we expand the exponential in spherical harmonics, use their orthogonality and obtain the decay amplitude as a sum of partial waves

(3)

where

(4)

and is a spherical Bessel function.

The total decay width obtained after integration over angles in this case is

(5)

In the calculations below the interaction potential is often a tensor, so the matrix elements similar to are nondiagonal in , in other words orbital momentum changes in decays. The total angular momentum with account for spins is of course conserved.

The effective potential

(6)

can be calculated in terms of the relativistic scattering amplitude with the nonrelativistic initial particles

(7)

The square roots in this relationship convert the relativistically normalized scattering amplitude to the normalization used in nonrelativistic quantum mechanics. It is convenient to rescale the potential so that it coincides with the amplitude

(8)

Then the total width in Eq. (5) acquires the form

(9)

Below we will use a natural generalization of this formula for particles with spin.

Our strategy is to use the standard Feynman rules with free initial and final particles to calculate the scattering amplitudes with the nonrelativistic initial particles. Then we convert the scattering amplitudes into effective potentials , expand the integrand in Eq. (2) in spherical harmonics (with account for spin, if necessary), calculate the angular integrals analytically and finish with computing the remaining radial integrals numerically, using the wave functions obtained in epp2018 ().

Let us illustrate the logic of calculations still assuming that all particles in Fig. 1 are scalars. In this case the rescaled potential is just

(10)

All external momenta are on mass shell and

(11)

and

(12)

In this simple case the potential is a function only of and its Fourier transform is just the Yukawa potential. Notice that its radius is determined not by the mass of the exchanged particle but by the effective mass .

ii.2 Tensor, Spin, and Isospin Structure of Decay Potentials

In the nonrelativistic approximation one-pion exchange in Fig. 1 generates a relatively long-range effective potential between and that was used in epp2018 () in discussion of the molecular pentaquark

(13)

where and are the axial charges of and , respectively, and matrix elements of the spin and isospin operators and should be calculated between the state vectors of the respective particles. In coordinate space the momentum-dependent factor turns into a superposition of a central and tensor potentials (we temporarily omit the coupling constants)

(14)

where and

(15)

There is also an additional term proportional to on the right hand side in Eq. (14) but we omit it as unphysical in calculations of the bound state energies, see epp2018 () for details. The spin and isospin matrices in Eq. (13) act in the space of spin and isospin states of the constituents. In epp2018 () we used the potentials in Eq. (13) and Eq. (15) together with the similar potentials that arise from , , and exchanges to construct a loosely bound pentaquark state . All potentials were regularized at small distances about 0.15 fm, for details of the regularization see Eq.(31,32) in epp2018 ().

Decays of molecular pentaquarks without charm exchange can go via exchanges by a pion and other light mesons. We expect that the one-pion contribution, without account for exchanges by other mesons, gives a reasonable estimate of decay widths. Unlike the case of the binding potential, one-particle exchange decay amplitudes describe transitions from one pair of particles to another. After calculations pion exchange reduces to the potentials of the same type as in Eq. (14) and Eq. (15), the only differences are that we use the nondiagonal axial charges (see also hpkmw2017 ()), and substitute and , compare Eq. (11). Decays of the molecular and hadrocharmonium pentaquarks with exchange of charm go via -meson and other heavy hadron exchanges. The respective effective potentials do not coincide with the ones in Eq. (14) and Eq. (15), but still depend on spin, isospin and orbital momenta. This allows us to give a universal description of the strategy of further calculations. Consider, for example, a molecular pentaquark decay. The bound state wave function of the molecular pentaquark epp2018 () is a superpositions of the states , , and , where is the orbital momentum and is the total spin of the pentaquark. Each of the components of the molecular wave function is in its turn a superposition of one-particle spin-isospin states of the constituents. In terms of these spin-isospin states of the constituents the the wave function of the pentaquark in the state has the form

(16)

where and are normalized to unity spin-isospin states of and with the spin projection and the isospin projection , are the third components of the pentaquark spin and isospin, are spherical harmonics, , , are the Clebsch-Gordan coefficients, and are the radial wave functions in the states . Summation runs over spin and isospin projections of the constituents and includes also summation over three available combinations.

We consider a one-particle exchange scattering amplitude as an operator that acts on the initial wave function in Eq. (16) and transforms it in a superpositions of products of spin-isospin one-particle states of the final particles with the coefficients that are coordinate wave functions of their relative motion. Like in Eq. (16) these coordinate wave functions are themselves superpositions of products of radial wave functions and spherical harmonics. The final orbital momenta arise automatically by addition of orbital momenta of the initial wave function and of the interaction potential and do not coincide with the initial orbital momenta, only the total angular momentum is conserved in the general case. Next we project this wave function on the final plane wave, compare Eq. (3). We obtain a superposition of matrix elements of the potential (compare Eq. (4)), with the coefficients that are spin-isospin wave functions of the final particles. Unlike the expression in Eq. (4) the radial wave function carries now a second index because it depends on the total spin of the bound state. In addition the final angular momentum in the integral for does not necessarily coincide with the initial angular momentum since the potential is in the general case a coordinate space (as well as spin and isospin) tensor. These matrix elements are decay amplitudes of the initial state into a final state with the total orbital momentum and spin-isospin quantum numbers of the coefficients.

To calculate the decay width in any channel we apply the operator arising from the respective one-particle exchange amplitude to the wave function Eq. (16) of the pentaquark with fixed quantum numbers. Then we obtain the decay amplitude as a superposition of matrix elements , square it, calculate the integrals over directions of the final momentum and thus obtain the decay width. We will fill some technical gaps in this schematical discussion considering the decays below.

Iii Decays of Molecular Pentaquarks

Let us recall the principal features of the molecular pentaquark scenario considered in epp2018 (). Exotic pentaquarks in this picture are loosely bound states of hadrons with open charm located at rather large distances. One could expect that the interaction of the constituent hadrons in this case would be dominated by the long-range one-pion exchange and the pentaquark would resemble the deuteron, see, e.g., nat1991 (). We considered this binding mechanism in epp2018 () and came to the conclusion that the effective distances are not large enough to neglect exchanges by other light mesons, besides pions. The pion exchange in epp2018 () was regularized to get rid of its unphysical too singular behavior at small distances, and exchanges by , , and were also taken into account. Then we constructed the pentaquark as a loosely bound state of () and () with the binding energy only 15 MeV and spin-parity . This pentaquark arises when the regularization parameter MeV, with the root mean square radius fm and -wave squared fraction about 4%, see epp2018 () for more details. An attempt to use the potential with the same parameters in order to construct as a loosely bound state of () and () with the binding energy 10 MeV was not successful. The main reason is that the would be constituents and do not interact via one-pion exchange since the three-pseudoscalar vertex is banned by parity, and exchanges by the other light mesons cannot provide the necessary binding. Therefore, if we insist that the LHCb pentaquark should be a loosely bound molecular state with a tiny binding energy its nature in this picture remains an open question.

Small binding energy and large size of the molecular pentaquark imply that the constituent hadrons are non-relativistic and this bound state can be described in the potential approach. We constructed such molecular pentaquark in epp2018 (). Let us consider its decays due to one-particle exchanges.

iii.1 Decays into States with Open Charm

There are four open channels for the pentaquark decays into states with open charm, see Table 1. In the case of the molecular pentaquark there is no charm exchange in these decays and they can go via one-pion exchanges. As mentioned above, exchanges by heavier mesons are also allowed but we will account only for the contribution of the pion exchange.

iii.1.1 Decay

Figure 2: Decay of molecular pentaquark into open charm states

We start with the channel . The initial pentaquark has spin-parity and isospin , the final carries spin-parity and zero isospin, and the final is a pseudoscalar with isospin . The product of the internal parities of and is negative, so the final state in the decay can have only even angular momenta. The final state with is banned by the angular momentum conservation, so the lowest allowed final orbital momentum is . The final decay momentum is MeV, and both final particles are nonrelativistic with a reasonable accuracy, MeV and , and MeV and .

This decay is described by the diagram in Fig. 2. First we calculate the relativistic scattering amplitude in Fig. 3

(17)

where is a four-vector isospinor, is an isospinor, is a spinor isovector, and is a spinor. The coupling constants and interaction Lagrangians can be found in Table 5 and are discussed in Appendix A.1.

Figure 3: Amplitude

In the nonrelativistic approximation the denominator of the propagator reduces to , and the interaction radius is determined by MeV. Using this approximation for the the initial and final particles and omitting the coupling constants and certain square roots of masses (to be restored in the final expression for the decay width, compare Eq. (8)) we obtain the interaction potential that acts as an operator on the initial pentaquark wave function in Eq. (16)

(18)

or in coordinate space

(19)

where is defined in Eq. (14) (now with ) and are nonrelativistic spin-isospin states similar to the ones in Eq. (16).

It is convenient to represent in terms of spherical harmonics333We use conventions for spherical harmonics from ll1991 (), in particular (20)

(21)

where and are the regularized potentials in Eq. (15), see discussion of the regularization below Eq. (15) and in epp2018 ().

The transition operator in Eq. (19) should be applied to the initial wave function of the molecular pentaquark. We choose the initial pentaquark state with and . The interaction operator in Eq. (19) transforms it into the final wave function. After projection on the final plane wave and spatial integration we obtain the decay amplitude

(22)

where is the final with spin up or down, , and are radial matrix elements of the potentials between the initial pentaquark state and the final two-particle state with the orbital momentum similar to the ones in Eq. (4). We see that interaction in Eq. (19) generates only the transitions to the final states in -wave. Next we calculate module square of the transition matrix element in Eq. (22), integrate over the directions of the final momentum, and sum over all allowed final states

(23)

The decay width is calculated with a natural generalization of Eq. (9)

(24)

where we plug in , , , , , , , , and sum of matrix elements squared from Eq. (23). We use Eq. (24) for calculations of all decay widths below.

After numerical calculations we obtain MeV.

iii.1.2 Other Open Charm Decays of Molecular Pentaquark

Calculation of other three decays of the molecular pentaquark into states with an open charm

(25)

is similar to the calculations above. All these decays go via the pion exchange, the final decay momenta are even smaller than in the decay , see Table 1, and the decay products are nonrelativistic.

Decay requires almost no new calculations. Spin-parity of are the same as spin-parity of and like in the previous decay is the lowest allowed partial wave. The final momentum is MeV, and the final particles are again essentially nonrelativistic. Kinetic energy of the -meson is about 4% of its mass, and kinetic energy of is about 2% of its mass.

The decay amplitude in Fig. 4 can be obtained from the decay amplitude in Fig. 2. Only the isotopic structure of the vertex is different from the isotopic structure of the vertex, see the respective interaction Lagrangians in Table 5. The isotopic factor factorizes in the decays amplitudes and the decay width of is equal to the decay width of times the ratio of the respective isotopic factors squared.

Figure 4: Decay of molecular pentaquark into open charm states

The isospinor isotopic factor in the molecular pentaquark wave function is . In the case of decay we apply to this wave function the isotopic factor in the transition operator in Eq. (19) and obtain the final isotopic function

(26)

In the case of the decay the isotopic factor in the transition operator in the diagram in Fig. 4 is and then the final isotopic wave function is

(27)

Squaring the isotopic factors in the scattering amplitudes and summing over all allowed final isotopic states we obtain the isotopic factor contributions to the decay width in both cases

(28)

Spin and orbital structure of the matrix elements is identical for both decays. Hence, the sum of matrix elements squared for the decay is times larger than the sum of matrix elements squared for the decay , and (compare Eq. (23))

(29)

for .

Calculating the width according to Eq. (24) we obtain MeV.

The decay goes via the one-pion exchange diagram in Fig. 5. The interaction Lagrangian and coupling constant are in Table 5. Let us notice that both interaction constants in this decay are found from the experimental data on decays, see discussion in Appendix A.1.

Figure 5: Decay of molecular pentaquark into open charm states

We go through by now the standard steps and obtain a rather cumbersome sum of matrix elements squared for this decay

(30)

This sum is dominated by the second term that describes transitions between the states with zero orbital momentum. We substitute this sum in Eq. (24) and obtain MeV.

Figure 6: Decay of molecular pentaquark into open charm states

The decay goes via the one-pion exchange diagram in Fig. 6. The interaction Lagrangian and coupling constant are in Table 5. After calculations we obtain the sum of matrix elements squared

(31)

substitute it in Eq. (24) and calculate the width MeV.

iii.2 Decays into States with Hidden Charm

The decay is the only one kinematically allowed two-particle decay of the pentaquark into states with hidden charm. This decay goes via diagrams with exchange by a charmed meson or baryon in -channel, e.g., , , , etc. We will account only for the contribution of the diagram in Fig. 7 with the exchange by the lightest charmed particle, the pseudoscalar , that we expect to provide a reasonable estimate of the total decay width. The product of internal parities of and is negative, so decay goes with the lowest orbital momenta . The decay momentum MeV in this decay is comparable with the nucleon mass and one cannot use the nonrelativistic approximation for the final nucleon.

Figure 7: Decays of the molecular pentaquark into hidden charm states

As with the pion exchanges above, we start with calculation of the relativistic scattering amplitude in Fig. 8

(32)

where is a four-vector isospinor, is a spinor isovector, is a spinor isospinor, and is the polarization vector of the final . The coupling constants and interaction Lagrangians can be found in Table 6 and Table 7, and are discussed in Appendices A.2 and A.3.

Figure 8: Amplitude

Next we would like to make a nonrelativistic expansion in the initial momentum . The denominator of the propagator in Eq. (32) reduces to and the range of the effective potential is determined by MeV (). This effective potential acts at shorter distances than in the case of the molecular pentaquark decays into states with open charm. The zero component of the transferred momentum MeV is also large. Hence, we cannot neglect the decay momentum and zero component of the transferred momentum in the nonrelativistic limit. As a result the coordinate-dependent term in the transition operator

(33)

is more complicated than the similar term from Eq. (14) in a fully nonrelativistic case in Eq. (19). In the case at hand

(34)

The derivatives originate from the linear in the relative momentum terms in the numerator of the momentum space expressions. Due to these derivatives a new potential

(35)

arises in in Eq. (34) besides the potentials and from Eq. (15) ( plays the role of the mass parameter in all three potentials). We also keep the last bilinear in the final momentum term in the square brackets in Eq. (34) that cannot be legitimately omitted when the final momentum is large. All these new terms are missing in the nonrelativistic decays with exchange by an almost massless pseudogoldstone pion, because its interaction vertex is always proportional to its momentum. But nothing bans such interaction terms for a heavy .

The coefficients in Eq. (34) are functions of masses and the final momentum

(36)

where is the energy of the produced . Notice that these coefficients would be zero if masses of the constituent and the produced nucleon were close.

Further calculations go almost as in the case of the nonrelativistic decays above. A new element is connected with the scalar products like () that arise after differentiation in Eq. (34). We write them in terms of spherical harmonics , where are spherical components of . After application of the transition operator the final wave function contains products of different spherical harmonics that depend on and we use the Clebsch-Gordan coefficients to obtain terms linear in spherical harmonics, integrate over angles with the outgoing plane wave and obtain typical terms . Unlike the decays considered above, now such terms are multiplied by linear in the spherical components of factors. We calculate the radial integrals, project each of the products of spherical harmonics of on a single spherical harmonic , square the obtained sums and integrate over directions of . Notice that this calculation leads to the decay products with a final orbital momentum in ( is the label of the spherical Bessel function in the respective radial integral). The expression for the sum of matrix elements squared turns out to be rather cumbersome. The dominant contribution to this sum is supplied by the transitions from the component of the initial bound state wave function with , that has the form

(37)

where we introduced matrix element of a new type

(38)

that arises only for the odd values of . The potential in this integral is regularized in the same way as the potentials and in Eq. (15).

The final nucleon is relativistic in this decay and the general formula for the width in Eq. (24) changes

(39)

After numerical calculations we obtain decay width of the molecular pentaquark into states with hidden charm