Recoupling mechanism for exotic mesons and baryons

The infinite chain of transitions of one pair of mesons (channel I) into another pair of mesons (channel II) can produce bound states and resonances in both channels even if no interactions inside channels exist. These resonances which can occur also in meson-baryon channels are called channel-coupling (CC) resonances. A new mechanism of CC resonances is proposed where transitions occur due to a rearrangement of confining strings inside each channel — the recoupling mechanism. The amplitude of this recoupling mechanism is expressed via overlap integrals of the wave functions of participating mesons (baryons). The explicit calculation with the known wave functions yields the peak at E = 4.12 GeV for the transitions J/ψ+ϕ↔Ds∗+D¯s∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ J/\psi +\phi \leftrightarrow {D}_s^{\ast }+{\overline{D}}_s^{\ast } $$\end{document}, which can be associated with χc1 (4140), and a narrow peak at 3.98 GeV with the width 10 MeV for the transitions Ds−+D0∗↔J/ψ+K∗−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{-}+{D}_0^{\ast}\leftrightarrow J/\psi +{K}^{\ast -} $$\end{document}, which can be associated with th recently discovered Zcs (3985).


Introduction
The modern situation with the spectra of quarkonia and baryons requires the dynamical explanation of numerous extra states, which are not present in the one-channel spectra of a given meson [1]. 1 A similar situation occurs in the excited baryon spectra. 2 To be more precise, in the case of heavy quarkonia, i.e. states, which contain cc and bb pairs, the experimental data contain a number of charged Z c , Z b and neutral Y c , Y b states, which cannot be explained by the dynamics of cc, or bb pairs alone, see [2] for review.
There are theoretical suggestions of different mechanisms , which should be taken into account. E.g. poles (resonances) in the meson-meson channels can occur due to strong interaction in these systems, and appear as additional poles in the S matrix [4,7,8,18,19] -the molecular-type approach.

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A similar in the choice of the driving channels (QqQq), but different in the dynamics, is the approach of the tetraquark model [5, 10, 12-14, 17, 20, 24], see [25][26][27] for reviews. A more general approach contains features of both molecular and tetraquark models -the hybridized tetraquark model [28]. One of the basic features of these models (as well as of the hadron-hadron interaction in general) is the determination of the dynamics in the system of two white objects. In the case of a deep tetraquark state one can disregard the twobody white asymptotics with some accuracy,however in the molecular case and in general one needs a classification of possible dynamical exchanges V hh between two white hadrons, taking into account the full gauge invariance and the confinement via general Wilson loop representation, which is now standard. It is clear that one-gluon exchange and its spin versions (spin-spin,spin-orbit,tensor) are not present in V hh , while two-gluon exchange is the glueball exchange. The "exchange of confining interaction" is not a local potential but the transition of two white objects into a common white one with qq annihilations before and after, e.g. π + π → ρ → π + π. In a similar way one obtains (actually nonlocal) t−channel and u−channel hadron exchanges. A special role is played by the Coulomb (not color Coulomb) interaction, which always exists locally in addition to listed above interactions. It is important that all types of interaction between white objects (except for Coulomb) are short ranged for massive exchanged hadrons and do not produce narrow singularities by themselves. Our purpose in this paper is to find the dynamical origin of relatively narrow peaks nearby threshold which occur due to transformation of one white pair into another arbitrary number of times and to find what is the transformation itself. As we shall see, we shall meet a completely different type of transformation -we shall call it the recoupling and explain this mechanism in some detail. As a result we shall have here a new theoretical treatment of hadron-hadron interaction, suggesting a simple and quite general mechanism for exotic peaks in mesons and baryons -the recoupling mechanism.
In contrast to the approaches, where white-white interaction in the one-channel system, (e.g. in meson-meson) is generating resonances, we propose the dynamical picture, where the summed up transitions from one channel to another (without interaction inside channels) can be strong enough to produce resonances nearby thresholds. The specific feature of this interaction is that it depends strongly on the wave functions of both channels, entering in the overlap integral of the transition matrix element, which measures the amplitude of the transition between the initial qQ +qQ and final qq + QQ states.
In what follows we are exploiting the channel-coupling (CC) interaction in the form of the energy-dependent recoupling Green's functions as a possible origin of extra statesthe recoupling mechanism.
Indeed, more than 30 years ago, the present author participated in the systematic study of CC effects in the spectra of hadrons, nuclei and atoms [29]. It was found there, that the CC interaction defined by the Transition Matrix Element (TME) is able to produce resonances (poles) of its own, if TME is strong enough, i.e. if the corresponding TME satisfies certain conditions, similar to that for one-channel potential.
We show below, that at the basis of this recoupling process lies a simple picture of the string recoupling between the same systems of quarks and antiquarks, which does not need neither energy nor additional interaction, and is simply a kind of topological transformation of two confining strings with fixed ends into another pair of strings -the string recoupling.  One can see in figure 1 the confining regions (the crossed areas) for the bound states of quark-antiquark (mesons) q 1q2 and q 3q4 in the l.h.s. of the figure 1, which is transformed in the middle part of figure 1 into the confining region between q 1q4 on the plane of the figure, and "the confining bridge" -the double-crossed area betweenq 2 q 3 . The r.h.s. of the figure 1 is the same as the l.h.s. As the result the transition is (q 1q2 )+(q 3q4 ) ↔ (q 1q4 )+(q 2 q 3 ). It is interesting to understand what kind of vertices are responsible for this transition, and to this end we demonstrate in figure 2 below the possible construction of the "confining bridge" in the figure 1 by cutting the confining film and turning up the middle piece.

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We show in this way that topologically this process is equivalent to the double string breaking, and numerically is defined by the everlap integral of participating hadron wave functions. This mechanism is quite general and can work for meson-meson, meson-baryon, baryon-baryon states. In particular it can work for some of X, Y and Z states of heavy quarkonia, like Z c (3900) and Z c (4020).
It is a purpose of the present paper to exploit this formalism for the case of extra states in meson-meson or meson-baryon spectra and define possible resonances and thresholds, and further on to apply this formalism to the case of pentaquark states like P c (4312), P c (4440), P c (4457).
Our main procedure will be the calculation of TME using realistic wave functions of cc, bb, cū systems, as well as approximate for baryon systems. Using those we calculate the JHEP04(2021)051 resulting Green's functions and resonance positions and compare them with experiment. The plan of the paper is as follows.
In the next section we introduce the reader to the method by solving a simplified twochannel problem with a separable potential. Section 3 is devoted to the explicit formulation of the recoupling mechanism, section 4 contains application to the meson-meson channel, and section 5 considers the meson-baryon case. Section 6, 7 are devoted to the analysis of the physical structure and numerical results and discussion, while section 7 contains conclusions and an outlook.

The simplest case: only separable CC interaction
Suppose we have two channels 1 and 2 with thresholds E 1 and E 2 and the CC interaction is separable The Schroedinger-like (possibly relativistic) equations are and can be reduced to the equation Solving (2.3) one obtains the equation for the eigenvalue E For E 1 < E 2 one can put E = E 1 and get a condition for the existence of a bound state in our two-channel system [29].
One of the intriguing points now is how the bound state poles, or more generally, any poles appear when the interaction strength λ is large enough. To this end we make a simplifying assumption about the form of v i (k) and write where ν 1 , ν 2 and β 1 , β 2 are some constants. Assuming also the nonrelativistic kinematics 2µ i one obtains in the case a) We shall be mostly interested in the poles around the threshold E 2 and therefore in the first approximation we replace the first factor on the l.h.s. of (2.10) by a constant, assuming, that E 2 − E 1 has a large positive value, hence one can write for k 2 = µ 2 (E − E 2 ) using (2.10) . (2.11) From (2.1) one can see that the pole is originally (at λ ′ = 0) on the second E 2 sheet, k 2 = −iν 2 and remains on the second E 2 sheet with increasing λ ′ . Note, however, that since originally we have been on the E 1 first sheet, then Im λ ′ 2 > 0, and therefore Re k 2 < 0, implying that the pole can be of the Breit-Winger type for Re λ ′ 2 > ν 2 .
As will be shown below in section 4, resonance production cross sections are proportional to the function (2.12) We can generalize this separable form to the relativistic case, when two hadrons with masses m 3 , m 4 , so that the denominators in (2.5) look as follows: (2.13) Here we have two thresholds m 1 +m 2 and m 3 +m 4 , and we shall assume that m 1 +m 2 < m 3 + m 4 .
Making the replacement (2.13) in I 1 (E), I 2 (E) one can calculate these functions and find the behaviour of the approximate cross section in (2.12).

Equations for two channel amplitudes in the recoupling formalism
In this section we discuss the Green's function of the system of two white (noninteracting) hadrons h 1 , h 2 , which can transform into another system of white hadrons H 3 , H 4 and this transformation can occur infinite number of times , and the corresponding Green's functions as G h , G H , we obtain the total Green's function G αβ , e.g. G hh

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as a result one obtains the equation, which defines all possible singularities of the physical amplitudes, including the resonance poles.
Note, that the described above method of the channel coupling was proposed before in the nonrelativistic form by the Cornell group [30][31][32][33], and exploited for the analytic calculation of the charmonium spectra, where the h 1 h 2 are strongly interacting quarks cc. The subsequent development of this method in [34][35][36] has allowed to understand the nature of the X(3872) [35] and Z b states [36]. For the light quarks this method requires the explicit knowledge of qq spectrum and wave functions, which are available in the QCD string approach [37][38][39][40][41][42][43][44][45][46].
Recently the same approach, called the relativistic Cornell-type formalism successfully explained the spectrum of light scalars [47,48]. In our present case we disregard the interaction of hadrons h 1 with h 2 and H 3 with H 4 .
Both Green's function G H , G h describe propagation of two noninteracting subsystems, but each of these hadrons can have its own nontrivial spectrum.
In the simplest case, e.g. h = ππ, H = KK, the Green's functions of noninteracting particles are well known, see e.g. [47,48] for the scalar ππ, KK Green's functions with the fixed spatial distance between ππ or KK, needed to define the transition matrix element.
Since each of h i or H j is a composite system consisting of qq or qqq one must write the corresponding relativistic composite Green's function, using the path integral formalism, see [44][45][46] for a recent review.
As it is seen from (2.13), one needs the explicit form of the relativistic Green's function, consisting of two quark-antiquark mesons h 1 (q 1 ,Q 1 ) and h 2 (q 2 , Q 2 ) with the zero total momentum P = 0, so that the c.m. momentum of q 1Q1 is p 1 , while forq 2 Q 2 it is −p 1 . As a result the wave function of the h 1 h 2 system with P = 0 and c.m. coordinates R can be written as At the same time the relativistic wave function of the hadrons Here we have introduced the c.m. coordinates R of the hadrons, expressed via the average energies ω i , Ω i of the quarks and antiquarks in the hadron [49] Here ω i , Ω i are given in the appendix A of this paper.

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Next we must calculate the overlap matrix element of Introducing the Fourier component of the wave functions e.g.
The transition element (3.7) with the factorȳ 1234 , responsible for the recoupling of hadrons, shown in figure 1, has a simple structure. Indeed, as one can see in figure 1, the creation of two string configurations in the intermediate confining strings position and back into the original configuration. One may wonder what is the explicit mechanism of this recoupling, and what are the vertices denoted by thick points in the figure 2. To this end we note, that we have two strings on the r.h.s. of figure 2: string from Q 1 toQ 2 and another fromq 1 to q 2 ; this position results from the double string decay (the l.h.s. of figure 2) with the subsequent rotation of the string between Q 1 andQ 2 to the right, where this string is at some distance above the string betweenq 1 , q 2 . One can associate the quantity M (x, y) with this process and we must add this factor to V 12|34 . Writing M (x, y) = σ|x − y| in analogy with the one-point string decay described by the effective Lagrangian [50] for the string decay, and replacing it with the numerical value M ω , similarly to [34][35][36], (see appendix B for details) one can write y 1234 = M ω χ 1234 in (3.8), with χ 1234 describing the spin-isospin recoupling. Finally one obtains the expression for the whole combination .
The resulting singularities (square root threshold singularities and possible poles from the equation N (E) = 1) can be found in the integral (3.11).
One can see, that the structure of the expression (3.11) is the same as in eq. (2.5), provided V 12|34 factorizes in factors v 1 (p 1 )v 2 (p 2 ), and consequently one expects the same behaviour of the cross sections as in (2.12).
At this point it is useful to introduce the approximate form of the wave functions in (3.8), which is discussed in [36]. Here we only give the simplest form of the Gaussian wave functions for the ground states of light, heavy-light and heavy quarkonia. One can writeψ where β i was found in [34][35][36], see appendix C, e.g. for ground states of bottomonium β = 1.27 GeV, for charmonium β = 0.7 GeV and for D, B mesons β = 0.48, 0.49 GeV. Insertingψ i (q) in (3.12) into (3.8) and integrating over d 3 q 1 one obtains The resulting N (E) has the form (3.17) and the differential cross section with the final second channel is proportional to It is interesting, that for the fully symmetric case, when all β i are equal, and ω 1 = Ω 1 , one obtains for the exponent in (3.13) exp −

Recoupling mechanism for the meson-meson amplitudes
The formalism introduced on the previous section can be directly applied to the amplitudes, containing two meson-meson thresholds, m 1 + m 2 ↔ m 3 + m 4 with the singularities given by the equation As we discussed in section II, the conditions for the appearance of visible singularities require that the threshold difference ∆M = m 3 + m 4 − m 1 − m 2 should be comparable or smaller than average size β of the hadron wave functions in momentum space, while JHEP04(2021)051 the recoupling coefficientȳ 2 1234 is of the order of unity, i.e. there should be no angular momentum excitation or spin flip process.
An additional requirement is the relatively small widths of participating hadrons, otherwise all singularities would be smoothed out.
One can choose several examples in this respect. 2) One of the best studied exotic resonances Z c (3900) [55][56][57][58] was found in the reaction e + e − → π + π − J/ψ → π ± Z c (3900). It can be associated with the recoupling process DD * ↔ πJ/ψ, where the higher threshold is M 2 = 3874 MeV, and the spin, charge and isospin recombination agrees with this recoupling. One expects the peak above M 2 in agreement with experiment.

1) The set of tranformations
A similar situation can be in the case of the Z c (4020) observed in the reaction e + e − → ππh c [59], which can be associated with the recoupling πh c → D * D * with threshold Note, that in general the recoupling can easily produce both Z b , Z c resonance peaks, when a charged particle (like ρ) is participating in the sequence of transformations.

Recoupling mechanism for meson-baryon systems
One can consider the transformation sequence for baryons of the form, e.g.
In principle it implies the new degrees of freedom, associated with the additional quark in (qqq) as compared to the meson (qq). To simplify the matter, we start below with the assumption, that the diquark combination can be factorized out in the baryon (qqq) → q(qq) and does not change during the recoupling process, which can now be written as q(qq) + (QQ) ↔ Q(qq) + (qQ). (5.2)

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In doing so we neglect also the internal structure of the diquark (qq) system, which stays unchanged during the recoupling process, so that only its total spin, spin projection and its relative motion with the quark q orQ in the baryon bound state is present in the matrix element (3.11), while the norm of (qq) is factored out. As a result one can use eqs. (3.11), (3.13), where we need the wave functions of the relative motion of quark and diquark in the baryons q(qq) and Q(qq). Using our notations we are replacingq 2 by the diquark (qq ′ ). The accuracy of this replacement was discussed in literature [61][62][63][64][65] and the interactions are discussed and compared in [66]. In what follows we need the approximate baryon wave functions as where v denotes the center-of-mass of the diquark, and R(y, v) is the c.m. of the quark-diquark combination, i.e. actually is the c.m. of the baryon Q 2 (qq). The same for the ψ 3 (u − v) and its c. m. R(u, v). Using the oscillator forms for ψ 2 , ψ 3 one is actually exploiting the description of the only one part (factor) of the baryon wave function, which can be associated with only one leaf of the three-leaf baryon configuration. As a result, one can approximate this part of the wave function with the wave function of the heavy-light meson for the Q 2 (qq) baryon (Q 2 (qq) → Q 2q ) or with the light baryon for the h 3 (q 1q2 → q 1 (qq)).
As a first example one can take the transitions p + φ →K * + Λ with thresholds M 1 = 1960 MeV and M 2 = 2005 MeV, where the role of quarksQ 1 , Q 2 is played bys, s and one has a transition u(ud) +ss →su + s(ud), where all β parameters have a similar magnitude, and one can expect a peak nearby M 2 .
The most part of the literature considers pentaquarks as a result of molecular interaction between a white baryon and a white meson, which creates a bound state nearby the threshold of this system. In what follows we shall exploit the recoupling mechanism and we shall show, that it can provide the observed peaks without an assumption of the white-white strong interaction. We turn now to the recoupling coefficients M ω ,ȳ 1234 .
As it was shown in [50], the effective parameter M ω can be expressed via the wave functions of objects, produced by the string breaking, in our case it is heavy-light mesons with the coefficient β(D) ∼ = β(B) = 0.48 GeV, and from eq. (35) of [50] one has Finally, the coefficientχ 1234 for the transition into (Σ cD ) and (Σ cD * ) can be estimated as in the appendix B to be equal to 1.

Physical amplitudes and their singularities in the recoupling process
As was discussed in section 3, (3.18), the differential cross section for the production of hadrons in channel 1 can be written as

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where F 1 (E) is the production amplitude of channel 1 particles without final state F S interaction and f 12 is the F S interaction,which we take as an infinite sum of transitions from channel 1 to channel 2-the Cornell-type mechanism [30][31][32][33][34][35]. f 12 (E) can be written as where I i (E) has the form Here v i is proportional to the product of wave functions in momentum space (see (ref 42)) and can be written in two forms: a/ as a Gaussian of p and b/ as an inverse of (p 2 + ν 2 ).
To simplify matter we shall consider situation close to nonrelativistic for the energies in the denominator of (6.4) and write We have considered above in the paper v 2 (3.17). We simplify below this expression, writing . As a result one can write for I i (E) in the region E > E i (th) = m 1 + m 2 (i = 1) or m 3 + m 4 (i = 2) As a result one obtains a simple expression for the amplitude f 12 (E) .
We now consider the channel coupling constant λ ′ , which enters (6.7). From (3.17) one obtains The resulting z is larger than unity for the recoupling coefficient χ 1234 of the order of 1, and one can vary z in the interval from one to larger values. At this point we shall discuss the physical structure of the obtained expression for the amplitude f 12 (E) (6.7). One can see that f 12 contains the product of two unitary single channel amplitudes t 1 , t 2 which describe separately scattering amplitudes in channel 1 and 2 characterizing by numbers ν i , i = 1, 2. The latter are obtained from the hadron wave functions and reflect their JHEP04(2021)051 structure given by the Gaussian coefficients β i . Now let us study the structure of the possible singularities of f 12 in the E plane, which is given by zeros of the denominator of f 12 in (6.7) and we concentrate on the k 1 plane while the second channel is characterized by the value of B(E) = 1 ν 2 −ik 2 and −ik 2 > 0 below higher threshold E 2 . As a result one obtains an equation for the pole of f 12 It is evident that originally (at zB = 0) virtual pole of k 1 is proceeding to the upper half plane of k 1 with increasing zB, and at some zB at appears at k 1 = 0 i.e. at the threshold E 1 , and at larger zB one obtains a real pole, i.e. the effect of recoupling with the channel 2 increases the attraction and creates a real pole from the original distant virtual pole. However for the physical amplitude t 1 (E) of the hadron-hadron interaction the virtual pole can be not a good approximation and more generally one should write for the amplitude , where a, b are positive numbers. This construction occurs from the t-channel exchange of the hadron with mass m. One can see from (6.9) that the logarithmic singularity is moving towards the threshold with the increasing z and can appear on the first sheet (E < E 1 ). We can now study the situation with the singularities associated with the threshold E 2 and find that here the situation is completely different because when E E 2 the momentum k 1 (E) as a rule is large and positive imaginary. As a result there is no pole solutions of the equation (6.7) for all values of E nearby the threshold E 2 and all resulting widths are of the order of parameters µ i , ν i , i.e. (0.5 − 1) GeV, and therefore cannot be seen in experiment. Nevertheless we give in the Apendix 4 full quartic equation and its analysis for the finding of resonance poles in the whole combination of k 1 , k 2 planes. As a special case one can consider the situation when the distance ∆ = E 2 − E 1 is small as compared with ν i , i = 1, 2. In this case the amplitude f 12 (E) written as . (6.10) One can see that for k 1 ≪ ν 1 and ν 2 z ν 1 one obtains a resonance pole slightly shifted above the threshold E 2 , however this situation is coincidental. Summarizing we expect that the resulting singularities of the recoupling amplitude are produced by the singularities of the lower threshold process (process 1 in our definition) shifted to the lower threshold due to recoupling interaction with the process 2). In the next section we shall study this effect numerically in real physical examples.

Numerical results and discussion
We can now consider 3 transitions, partly discussed above:  Table 2. The values of the |f 12 (E)| 2 near the channel thresholds for the transition 1).
As a special interesting case we consider below the recent experiment of BES III [87], Applying here our recoupling mechanism, shown in the figure 1, one easily finds that the second channel obtained from the first channel D − s D * by recoupling is the channel J/ψ + K * − which creates the chain of reactions possibly generating a peak in the system D − s D * or D * − s D, namely we consider as the third example We proceed now with the cases 1)-3) and insert the values of E i (th), µ i , ν i in (6.3), (6.7) and fixing the value of z one obtains the form of the recoupling amplitude shown in the tables 2-4 below.
For the case 1) the resulting values of |f 12 (E)| 2 can be seen in the table 2.
One can see in the table 2 a strong enhancement around E = 4.12 GeV with the width around 10 MeV which can be associated with the resonance χ c1 (4140) having the mass 4147 MeV and the width Γ = 22 MeV.
It is now interesting how these numerical data are explained by the exact solution of the equation (6.7), (6.9). From these equations one obtains the exact position of the pole k 1 = i0.068 GeV, E = E 1 − 0.003 GeV which is unphysical result due to too small values of ν 1 , and in a more realistic case of the virtual pole its position moves to k 1 = 0, E = E 1 for ν 1 = 0.84 GeV. Three more complex poles are O(1 GeV) far from the thresholds, as can be found from the solution of the quartic equation in appendix D. Looking at table 2 one can see a good agreement with this result. In a similar way we obtain the results for the J/ψ + p transitions of 2).   Table 4. The values of the transition probability as a function of energy in the transition 3).

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One can see in table 3 a strong peak near the lower threshold E = 3.94 GeV and it is easy to check that this peak is stable when one varies z in the region around z = 1, One can associate this peak with the virtual pole appearing at 50 MeV below the threshold E 1 = 3.94 GeV. The exact form of the solution for f 12 (E) near the peak has the form f 12 (E) = k 1 +i0.96 k 1 +i0. 28 . In this way our method can support the origin of the X(3915) state as due to the J/ψ + φ ↔ D s +D s transitions.
We come now to the recent interesting discovery of the new state Z cs (3985) [87], where we take for simplicity only the first chain denoted as the 3) above. Similarly to the previous cases one obtains One can see in table 4 a narrow peak with the summit at E = 3.975 GeV for z = 1 with the width around 10 MeV, which closely corresponds to the experimental data from [87] E = 3.982.5, Γ = 12.8 MeV. In our case the resonance parameters weakly depend on z. As seen in the table 4, for z = 1.5 the peak shifts down to E = 3.960 GeV with much larger width. Its form is exactly the same as in the previous two examples and corresponds to virtual pole almost exactly at the lower threshold. In this way we can explain the newly discovered resonance Z cs (3985) by the recoupling mechanism in the rescattering series of transitions

Conclusions and an outlook
As it was shown above, the new mechanism having the only parameter z is able to predict and explain the resonances in different systems, as it was shown above, and possibly in other systems which can transfer one into another via the recoupling of the confining strings. The necessary conditions for the realization of these transitions and the appearance of a resonance are connected to the value of the transition coefficient z, which should be of the order of unity or larger. Therefore the transition should be strong, i.e. without serious restructuring of the hadrons involved, since otherwise the transition will be strongly suppressed e.g. in the case when not only strings are recoupled, but also spins,orbital momenta, isospins should be exchanged. In any case the suggested mechanism provides an alternative to the popular tetra -and pentaquark mechanisms, which dominate in the literature. One should stress at this point, that the independent and objective checks, e.g.

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the lattice calculations do not give strong support for the molecular or tetraquark models and the existence of an independent mechanism is welcome. It is necessary that this method should be studied more carefully. As to the recoupling mechanism, it is strongly associated with the thresholds participating in the transitions, and the best situation for its application is when both thresholds are close by. In all 3 cases considered above the distance between threshold was less than 200 MeV, and in all cases one could see a strong enhancement in the transition coefficient and hence in the resulting cross section. The necessary improvements of the present study are 1) a more accurate calculation of the coefficient z (originally λ ′ ), and 2) the use of a more realistic Gaussian approximation for the wave functions instead of approximate ν i parametrizations to define f 12 with good accuracy in the future.

A The center-of-mass coordinates and average quark and antiquark energies in a hadron
Following [49] one can define the c.m. coordinate of a hadron consisting of a quark Q at the point x and an antiquarkq at the point u via average energies Ω and ω ofQ andq correspondingly as where p 2 Q + m 2 Q + p 2 q + m 2 q is the kinetic part of the Qq Hamiltonian in the so-called spinless Salpeter formalism or an equivalent form in the so-called einbein formalism. 3 As a result one obtains the following value of ω = Ω for qq mesons, shown in table 2.

B The channel-coupling coefficientȳ 1234
We discuss here two topics: 1) the problem of the double string decay vertex contribution to the recoupling coefficient M ω in (3.11), 2) the construction of the recoupling vertexȳ 1234 . We start with the topic 1), and following [50] define the relativistic expression for the string decay vertex as in (3.9), but without free parameters namely M (x) in (3.9) is As one can see in figure 2, in our case the structure of the recoupling process can be explained by the double string breaking, which we can write as a product  and one must take into account, that the energy minimum of the resulting broken string occurs when both time moments x 0 , y 0 of string breaking are equal. Indeed, taking the integral in (B.2) with account of the string action in the exponent of the path integral, which produces a factor on (B.2) r xy √ 2π, which denoted as M ω in (3.11). The resulting double string breaking action can be written as, In what follows one can estimate M ω (x, y) ≡ M ω in the same way, as it was done in [50], with the result M ω ≈ 2σ β , where β is the oscillator parameter for the (QQ) meson. We now turn to the point 2) above, the recoupling vertexȳ 1234 .
To defineȳ 1234 we notice that all 4 quarks q,q, Q,Q keep their identity and spin polarization during the whole process of transformations, provided we neglect the spin dependent corrections. This can be also seen in the structure of the recoupling process: in (B.2) one does not see spin dependence, and this means, that the spin projection of each quark or antiquark is kept unchanged during recoupling. As a result one can write the nonrelativistic spin part of the matrix element V 12/34 as λ are quark and antiquark spinors.
As was told above, due to the spin conservation in recoupling, the matrix element (B.4) should be proportional to δ 26  As a result one obtains ( In this case we write the baryon wave function as ψ B = u(ud) 0 , where the lower indices imply the total spin of the diquark (ud). In the simplest approximation one can approximate the proton as the quark-diquark combination p = u(ud) ∼ = ud, with the diquarkd kept unchanged d during recoupling.

C Oscillator parameters of hadron wave function
The oscillator parameters for the bottomonium, charmonium and B, D mesons have been obtained in [37][38][39][40][41][42][43][44][45][46], using the expansion of relativistic wave functions, obtained from the solutions of the relativistic string Hamiltonian [49], in the full set of the oscillator wave functions. As a result one obtains The accuracy of the oscillator one-term approximation can be judged by the relative value of the sum od squared coefficients of four higher term of expansion as compared to the square of the main term. This amounts to the accuracy of the order or less than 10% for lowest states of charmonia and bottomonia and few percent for D, B, ρ.

D Calculation of the cross sections |f ik | 2 and pole positions
As it is written in (6.7) the cross section is defined as dσ dE = |f ik | 2 where f ik = 1 1−zAB , and z is numerical parameter proportional to λ ′ , where for three numerical examples (1), (2), (3)

JHEP04(2021)051
in that section z = 1, 5, 1 and A = 1 ν 1 −ik 1 and B = 1 ν 2 −ik 2 . Moreover k i = 2µ i (E − E i ) and parameters µ i , ν i , E i are given in the section. As it is seen in tables 2, 3, 4 the cross sections have peaks and our purpose here is to calculate the positions of the poles both in E plane and in k i plane. To do this one starts with two equations One of possible strategy is to insert k 1 from the first equation into second equation and one obtains an equation of 4-th power for k 2 . Note that both k i are positive on the real axis for E > E i and positive imaginary below E i . Among many roots of eq. we need the closest to the real axis. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.