aa r X i v : . [ h e p - ph ] N ov Recoupling Mechanism for exotic mesons and baryons
Yu.A. Simonov ∗ NRC “Kurchatov Institute” – ITEP, B. Cheremushkinskaya 25, Moscow, 117259, Russia
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 insidechannels 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 occurdue to a rearrangement of confining strings inside each channel – the recoupling mechanism. Theamplitude of this recoupling mechanism is expressed via overlap integrals of the wave functions ofparticipating mesons (baryons). The explicit calculation with the known wave functions yields thepeak at E = 4 .
12 GeV for the transitions
J/ψ + φ ↔ D ∗ s + ¯ D ∗ s , which can be associated with χ c (4140), the peak at 4 . J/ψ + p ↔ Σ c + D ∗ , which can be associated with P c (4457), and a narrow peak at 3 .
98 GeV with the width 10 MeV for the transitions D − s + D ∗ ↔ J/ψ + K ∗− , which can be associated with th recently discovered Z cs (3985). PACS numbers:
I. INTRODUCTION
The modern situation with the spectra of quarkonia and baryons requires thedynamical explanation of numerous extra states, which are not present in the one-channel spectra of a given meson [1]. A similar situation occurs in the excitedbaryon spectra [2]. To be more precise, in the case of heavy quarkonia, i.e. states,which contain c ¯ c and b ¯ b 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 c ¯ c ,or b ¯ b pairs alone, see [3] for review.There are theoretical suggestions of different mechanisms [4–23], which should betaken into account. E.g. poles (resonances) in the meson-meson channels can occurdue to strong interaction in these systems, and appear as additional poles in the S matrix [5, 8, 17, 18] – the molecular-type approach.A similar in the choice of the driving channels ( Q ¯ q ¯ Qq ), but different in the dynam-ics, is the approach of the tetraquark model [6, 10, 12, 13, 16, 19, 23], see [24–26] forreviews. We shall consider here a different theoretical treatment of this problem,suggesting a simple and quite general mechanism for exotic peaks in mesons andbaryons – the recoupling mechanism.In contrast to the approaches, where white-white interaction in the one-channelsystem, (e.g. in meson-meson) is generating resonances, we propose the dynamicalpicture, where the summed up transitions from one channel to another (without ∗ Electronic address: [email protected] interaction inside channels) can be strong enough to produce resonances nearbythresholds. The specific feature of this interaction is that it depends strongly onthe wave functions of both channels, entering in the transition matrix element,which measures the amplitude of the transition between the initial q ¯ Q + ¯ qQ andfinal q ¯ q + Q ¯ Q states.In what follows we are exploiting the channel-coupling (CC) interaction in theform of the energy-dependent recoupling Green’s functions as a possible origin ofextra states - the recoupling mechanism.Indeed, more than 30 years ago, one of the present authors participated in thesystematic study of CC effects in the spectra of hadrons, nuclei and atoms [27]. Itwas 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-channelpotential.We show below, that at the basis of this recoupling process lies a simple pictureof the string recoupling between the same systems of quarks and antiquarks, whichdoes not need neither energy nor additional interaction, and is simply a kind oftopological transformation of two confining strings with fixed ends into anotherpair of strings – the string recoupling. FIG. 1: The transition of the mesons ( q ¯ q ) + ( q ¯ q ) ↔ ( q ¯ q ) + (¯ q q ) via recoupling of the confining strings. One can see in Fig. 1 the confining regions (the crossed areas) for the boundstates of quark-antiquark(mesons) q ¯ q and q ¯ q in the l.h.s. of the Fig. 1, whichis transformed in the middle part of Fig. 1 into the confining region between q ¯ q on the plane of the figure, and “the confining bridge” – the double-crossed areabetween ¯ q q . The r.h.s. of the Fig. 1 is the same as the l.h.s. As the result thetransition is ( q ¯ q ) + ( q ¯ q ) ↔ ( q ¯ q ) + ( ¯ q q ). It is interesting to understand what kind of vertices are responsible for this transition,and to this end we demonstratein Fig. 2 below the possible construction of the “confining bridge” in the Fig. 1 bycutting the confining film and turning up the middle piece. FIG. 2: The mechanism of string recoupling via double string-breaking process at the points shown by thick dots.
We show in this way that topologically this process is equivalent to the doublestring breaking, and numerically is defined by the everlap integral of participatinghadron 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 extrastates in meson-meson or meson-baryon spectra and define possible resonances andthresholds, and further on to apply this formalism to the case of pentaquark stateslike P c (4312) , P c (4440) , P c (4457).Our main procedure will be the calculation of TME using realistic wave functionsof c ¯ c , b ¯ b , c ¯ u systems, as well as approximate for baryon systems. Using those wecalculate the resulting Green’s functions and resonance positions and compare themwith experiment. The plan of the paper is as follows.In the next section we introduce the reader to the method by solving a simpli-fied two-channel problem with a separable potential. Section 3 is devoted to theexplicit formulation of the recoupling mechanism, section 4 contains application tothe meson-meson channel, and section 5 considers the meson-baryon case. Sec-tion 6 is devoted to the numerical results and discussion, while section 7 containsconclusions and an outlook. II. THE SIMPLEST CASE: ONLY SEPARABLE CC INTERACTION
Suppose we have two channels 1 and 2 with thresholds E and E and the CCinteraction is separable V ( p , p ) = − λv ( p ) v ( p ) = V . (1)The Schroedinger-like (possibly relativistic) equations are( T − E ) ϕ + V ϕ = 0 , ( T − E ) ϕ + V ϕ = 0 (2)and can be reduced to the equation( T − E ) ϕ + V ( E ) ϕ = 0 , (3)where V is V ( p , p ′ E ) = − λ v ( p ) v ( p ′ ) Z v ( k ) d k/ (2 π ) T ( k ) + E − E . (4)Solving (3) one obtains the equation for the eigenvalue E λ Z v ( p ) d p/ (2 π ) T ( p ) + E − E Z v ( k ) d k/ (2 π ) T ( k ) + E − E , (5)or 1 = λ I ( E ) I ( E ) . (6)For E < E one can put E = E and get a condition for the existence of a boundstate in our two-channel system [27]. λ I ( E ) I ( E ) ≥ . (7)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 wemake a simplifying assumption about the form of v i ( k ) and write a ) v i ( k ) = 1 k + ν i b ) v i ( k ) = exp (cid:18) − k β i (cid:19) (8)where ν , ν and β , β are some constants. Assuming also the nonrelativistic kine-matics T i = k µ i one obtains in the case a) I i ( E ) = µ i π ( ν i − i √ µ i ∆ i ) , ∆ i = E − E i , (9) and we have taken square root (9) on the physical Riemann sheet, E = E + iδ .Hence the equation (5) for the poles (energy eigenvalues) is( ν − i p µ ( E − E ))( ν − i p µ ( E − E )) = C = µ µ λ π . (10)We shall be mostly interested in the poles around the threshold E and thereforein the first approximation we replace the first factor on the l.h.s. of (10) by aconstant, assuming, that E − E has a large positive value, hence one can write for k = p µ ( E − E ) using (10) k ∼ = − iν + iλ ′ , λ ′ = µ µ λ π ( ν − i p µ ( E − E )) . (11)From (1) one can see that the pole is originally (at λ ′ = 0) on the second E sheet, k = − iν and remains on the second E sheet with increasing λ ′ . Note,however, that since originally we have been on the E first sheet, then Im λ ′ > k <
0, implying that the pole can be of the Breit-Winger type forRe λ ′ > ν .As will be shown below in section 4, resonance production cross sections are pro-portional to the function dσ ∗ ( E ) dE = (cid:12)(cid:12)(cid:12)(cid:12) − λ I ( E − E ) I ( E − E ) (cid:12)(cid:12)(cid:12)(cid:12) k ( E ) . (12)We can generalize this separable form to the relativistic case, when two hadronswith masses m , m , so that the denominators in (5) look as follows: T ( p ) + E − E → q p + m + q p + m − E,T ( p ) + E − E → q k + m + q k + m − E. (13)Here we have two thresholds m + m and m + m , and we shall assume that m + m < m + m .Making the replacement (13) in I ( E ) , I ( E ) one can calculate these functions andfind the behaviour of the approximate cross section in (12). III. EQUATIONS FOR TWO CHANNEL AMPLITUDES IN THE RECOUPLING FORMALISM
In this section we discuss the Green’s function of the system of two white (nonin-teracting) hadrons h , h , which can transform into another system of white hadrons H , H and this transformation can occur infinite number of times h h → H H → h h → H H → ... Denoting the transition amplitude V ( h h → H H ) = V + ( H H → h h ), andthe corresponding Green’s functions as G h , G H , we obtain the total Green’s function G αβ , e.g. G hh G hh = G h + G h V hH G H V Hh G h + G h V hH G H V Hh G h V hH G H V Hh G h + ... = G h − V hH G H V Hh G h ;(14)as a result one obtains the equation, which defines all possible singularities of thephysical amplitudes, including the resonance poles.1 = V hH G H V Hh G h , (15)Note, that the described above method of the channel coupling was proposedbefore in the nonrelativistic form by the Cornell group [28], and exploited for theanalytic calculationof the charmonium spectra, where the h h are strongly interact-ing quarks c ¯ c . The subsequent development of this method in [29–31] has allowed tounderstand the nature of the X (3872) [30] and Z b states [31]. For the light quarksthis method requires the explicit knowledge of q ¯ q spectrum and wave functions,which are available in the QCD string approach [32, 33].Recently the same approach, called the relativistic Cornell-type formalism suc-cessfully explained the spectrum of light scalars [34, 35]. In our present case wedisregard the interaction of hadrons h with h and H with H .Both Green’s function G H , G h describe propagation of two noninteracting subsys-tems, but each of these hadrons can have its own nontrivial spectrum.In the simplest case, e.g. h = π ¯ π, H = K ¯ K , the Green’s functions of noninter-acting particles are well known, see e.g. [34, 35] for the scalar ππ, K ¯ K Green’sfunctions with the fixed spatial distance between ππ or K ¯ K , needed to define thethe transition matrix element.Since each of h i or H j is a composite system consisting of q ¯ q or qqq one must writethe corresponding relativistic composite Green’s function, using the path integralformalism, see [33] for a recent review.As it is seen from (13), one needs the explicit form of the relativistic Green’sfunction, consisting of two quark-antiquark mesons h ( q , ¯ Q ) and h ( ¯ q , Q ) withthe zero total momentum P = 0, so that the c.m. momentum of q ¯ Q is p , whilefor ¯ q Q it is − p . As a result the wave function of the h h system with P = 0and c.m. coordinates R can be written as Ψ h h ( u − x ; y − v ) = e i p R ( u , x ) − i p R ( y , v ) ψ ( u − x ) ψ ( y − v ) (16)At the same time the relativistic wave function of the hadrons H , H , h H ( q ¯ q ) , H ( ¯ Q Q ) has the formΨ h h ( u − v ; v − y ) = e i p R ( u , v ) − i p R ( x , y ) ψ ( u − v ) ψ ( x − y ) . (17)Here we have introduced the c.m. coordinates R of the hadrons, expressed via theaverage energies ω i , Ω i of the quarks and antiquarks in the hadron [36] R ( u , x ) = ω u + ¯Ω x ω + ¯Ω , R ( y , v ) = ¯ ω v + Ω y ¯ ω + Ω , (18) R ( u , v ) = ω u + ¯ ω v ω + ¯ ω , R ( x , y ) = ¯Ω x + Ω y ¯Ω + Ω . (19)Here ω i , Ω i are given in the Appendix 1 of this paper.Next we must calculate the overlap matrix element of Ψ h h and Ψ h h V | ( p , p ) = Z ¯ y d ( u − x ) d ( y − v ) d ( u − v )Ψ h h Ψ + h h . (20)Introducing the Fourier component of the wave functions e.g. ψ ( u − x ) = R ˜ ψ ( q ) e i q ( u − x ) d q (2 π ) , one obtains in the simple case when q = q , Q = Q V | ( p , p ) = Z d q (2 π ) ¯ y ˜ ψ ( q ) ˜ ψ ( q + p ) ψ (cid:18) − q − p − p ω ω + Ω (cid:19) ×× ψ (cid:18) q − p − p Ω ω + Ω (cid:19) (21)In (20), (21) we introduced the numerical recoupling coefficient ¯ y , which isdiscussed in Appendix 2.The transition element (20) with the factor ¯ y , responsible for the recouplingof hadrons, shown in Fig. 1, has a simple structure. Indeed, as one can see in Fig.1, the creation of two string configurations in the intermediate confining stringsposition and back into the original configuration. One may wonder what is theexplicit mechanism of this recoupling, and what are the vertices denoted by thickpoints in the Fig. 2. To this end we note, that we have two strings on the r.h.s. of Fig. 2: string from Q to ¯ Q and another from ¯ q to q ; this position results fromthe double string decay (the l.h.s. of Fig. 2) with the subsequent rotation of thestring between Q and ¯ Q to the right, where this string is at some distance abovethe string between ¯ q , q . One can associate the quantity M ( x, y ) with this processand we must add this factor to V | . Writing M ( x, y ) = σ | x − y | in analogy withthe one-point string decay described by the effective Lagrangian [37] for the stringdecay, L sd = Z d x ¯ ψ ( x ) M ( x ) ψ ( x ) (22)and replacing it with the numerical value M ω , similarly to [29–31], (see Appendix2 for details) one can write y = M ω χ in (21), with χ describing the spin-isospin recoupling. Finally one obtains the expression for the whole combination N ( E ) = G h h V / G h h V / (23) N ( E ) = Z d p (2 π ) d p (2 π ) V | ( p , p ) V | ( p , p )( E ( p ) + E ( p ) − E )( E ( p ) + E ( p ) − E ) . (24)The resulting singularities (square root threshold singularities and possible polesfrom the equation N ( E ) = 1) can be found in the integral (24).One can see, that the structure of the expression (24) is the same as in Eq.(5),provided V | factorizes in factors v ( p ) v ( p ), and consequently one expects thesame behaviour of the cross sections as in (12).At this point it is useful to introduce the approximate form of the wave functions in(21), which is discussed in [31]. Here we only give the simplest form of the Gaussianwave functions for the ground states of light, heavy-light and heavy quarkonia. Onecan write ˜ ψ i ( q ) = c i exp (cid:18) − q β i (cid:19) , c i = 8 π / β i ; Z ˜ ψ i ( q ) d q (2 π ) = 1 (25)where β i was found in [29–31], see Appendix 3, e.g. for ground states of bot-tomonium β = 1 .
27 GeV, for charmonium β = 0 . D, B mesons β = 0 . , .
49 GeV.Inserting ˜ ψ i ( q ) in (25) into (21) and integrating over d q one obtains V | ( p p ) = ¯ y Y i =1 c i ! exp( − AP − Bp − C p p )(2 π ) π / a / , (26) where a, A, B, C are a = 12 X i =1 β i ; A = 12 (cid:18) β + 14 β + 14 β (cid:19) − a (cid:18) β + 12 β + 12 β (cid:19) (27) B = 12 β (cid:18) ω ω + Ω (cid:19) + 12 β (cid:18) Ω ω + Ω (cid:19) − a (cid:18) β ω ω + Ω − β Ω ω + Ω (cid:19) (28) C = 12 (cid:18) β ω ( ω + Ω ) − β Ω ( ω + Ω ) (cid:19) − a (cid:18) β + 12 β + 12 β (cid:19)(cid:18) ω β ( ω + Ω − Ω β ( ω + Ω (cid:19) . (29)The resulting N ( E ) has the form N ( E ) = M ω ¯ χ a ( Q i β i ) ( π ) Z d p d p exp( − Ap − Bp − C p p )( E ( p ) + E ( p ) − E )( E ( p ) + E ( p ) − E ) (30)and the differential cross section with the final second channel is proportional to dσdE ∼ p ( E ) | − N ( E ) | (31)where p ( E ) ∼ p E − ( m + m ) . It is interesting, that for the fully symmetriccase, when all β i are equal, and ω = Ω , one obtains for the exponent in (26)exp (cid:16) − p + p β (cid:17) , and V | = V | and N ( E ) are V symm12 | ( p , p ) = 2 / √ πβ ¯ y exp (cid:18) − p + p β (cid:19) (32) N ( E ) = 2 M ω ¯ χ πβ Z p dp p dp exp (cid:16) − p + p β (cid:17) ( E ( p ) + E ( p ) − E )( E ( p ) + E ( p ) − ( E ) (33) IV. RECOUPLING MECHANISM FOR THE MESON-MESON AMPLITUDES
The formalism introduced on the previous section can be directly applied to theamplitudes, containing two meson-meson thresholds, m + m ↔ m + m with thesingularities given by the equation1 − N ( m , m , m , m ; E ) = 0 . (34) As we discussed in section II, the conditions for the appearance of visible singu-larities require that the threshold difference ∆ M = m + m − m − m shouldbe comparable or smaller than average size h β i of the hadron wave functions inmomentum space, while the recoupling coefficient ¯ y 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. The set of tranformations
J/ψ + φ ↔ D ∗ s + ¯ D ∗ s → J/ψ + φ (35)with masses m = 3097 MeV, m = 1020 MeV, m = m = 2112 MeV, and thecorresponding thresholds m + m = 4117 MeV and m + m = 4224 Mev. Onecan see no spin flip in the sequence c + ¯ c + + s + ¯ s − → c + ¯ s − + ¯ c + s + for (34), wherelower indices denote spin projections, and therefore no damping of transitionprobability. One can expect, that the yield of the reaction (35) would havethe form similar to that of χ c (4140) with the mass (4147 MeV with the widthΓ = (22 ± M eV [38]. One of the best studied exotic resonances Z c (3900) [39–42] was found in the reac-tion e + e − → π + π − J/ψ → π ± Z c (3900). It can be associated with the recouplingprocess D ¯ D ∗ ↔ πJ/ψ , where the higher threshold is M = 3874 MeV, and thespin, charge and isospin recombination agrees with this recoupling. One expectsthe peak above M in agreement with experiment.A similar situation can be in the case of the Z c (4020) observed in the reaction e + e − → ππh c [43], which can be associated with the recoupling πh c → D ∗ ¯ D ∗ with threshold M = 3665 MeV and M = 4020 MeV. One can one can envisagethe yield of the reaction to be described by the equation (31), with p ( E ) → p ( E ), since one need the P -wave in D ∗ ¯ D ∗ near threshold, as in h c .Note, that in general the recoupling can easily produce both X, Y or Z c resonancepeaks, when a charged particle (like ρ ) is participating in the sequence of transfor-mations. V. RECOUPLING MECHANISM FOR MESON-BARYON SYSTEMS
One can consider the transformation sequence for baryons of the form, e.g.( qqq ) + Q ¯ Q ) ↔ ( qqQ ) + ( q ¯ Q ) (36) and apply the same formalism as the used above for the meson-meson recouplingtransformations.In principle it implies the new degrees of freedom, associated with the additionalquark in ( qqq ) as compared to the meson ( q ¯ q ). To simplify the matter, we startbelow with the assumption, that the diquark combination can be factorized out inthe baryon ( qqq ) → q ( qq ) and does not change during the recoupling process, whichcan now be written as q ( qq ) + ( Q ¯ Q ) ↔ Q ( qq ) + ( q ¯ Q ) . (37)In doing so we neglect also the internal structure of the diquark ( qq ) system, whichstays unchanged during the recoupling process, so that only its total spin, spinprojection and its relative motion with the quark q or Q in the baryon bound stateis present in the matrix element (24), while the norm of ( qq ) is factored out. Asa result one can use eqs. (24), (26), where we need the wave functions of therelative motion of quark and diquark in the baryons q ( qq ) and Q ( qq ). Using ournotations h ( q ¯ Q ) + h ( ¯ q Q ) ↔ h ( q ¯ q ) + h ( ¯ Q Q ) we are replacing ¯ q by thediquark ( qq ′ ). The accuracy of this replacement was discussed in literature [45–49] and the interactions are discussed and compared in [50]. In what follows weneed the approximate baryon wave functions as ψ ( y − v ) e − i p R ( y , v ) and ψ ( u − v ) e − i p R ( u , v ) in (15),(17), 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. ofthe baryon Q ( qq ). The same for the ψ ( u − v ) and its c.m. R ( u , v ). Using theoscillator forms for ψ , ψ one is actually exploiting the description of the only onepart (factor) of the baryon wave function, which can be associated with only oneleaf of the three-leaf baryon configuration. As a result, one can approximate thispart of the wave function with the wave function of the heavy-light meson for the Q ( qq ) baryon ( Q ( qq ) → Q ¯ q ) or with the light baryon for the h ( q ¯ q → q ( qq )).As a first example one can take the transitions p + φ → ¯ K ∗ + Λ with thresholds M = 1960 MeV and M = 2005 MeV, where the role of quarks ¯ Q , Q is played by¯ s, s and one has a transition u ( ud ) + ¯ ss → ¯ su + s ( ud ), where all β parameters havea similar magnitude, and one can expect a peak nearby M .As a concrete example one can take the case of the triad P c (4312) , P c (4440) and P c (4457), found experimentally in [51, 52], with a vast literature devoted to this phe-nomenon, called pentaquark, see a review in PDG [2], and for the latest pentaquarkpapers see [53–70] .The most part of the literature considers pentaquarks as a result of molecularinteraction between a white baryon and a white meson, which creates a bound statenearby 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 anassumption of the white-white strong interaction.We shall have in mind the recoupling transformations of the type J/ψ + P ↔ (Σ , Σ ∗ ) + ( D, D ∗ ) (38)and we impose the requirement of s -wave recoupling without spin flip processesand parity conservation, which excludes Λ ∗ c (2595) with ( I J P ) = (0 , − ) and in-cludes Λ c (2286)(0 ,
12 + ), Σ c (2455)(1 ,
12 + ) , Γ ≈ ∗ c (2529)(1 ,
32 + ) , Γ ≈
15 MeV,in addition to D (1864) , ( , + ) and D ∗ (2010) , ( , − ) with Γ D , Γ D ∗ < M = m + m in the Table 1 together with P c . TABLE I: Meson-baryon thresholds and the associated pentaquarksthresholds (MeV) 4150 4319 4465 4384pairs Λ c ¯ D Σ c ¯ D Σ c ¯ D ∗ Σ ∗ c ¯ D pentaquarks P c (4312) P c (4457) P c (4440)width, MeV Γ = 9 . . . Following the Table, we can consider two types of reactions,
I . ( c ( ud ))(Σ c , Σ ∗ c ) + (¯ cu )( ¯ D, ¯ D ∗ ) ↔ ( c ¯ c )( J/ψ ) + ( u ( ud ))( p ) (39) I I . ( c ( uu ))(Σ ∗ c ) + ¯ cd ( ¯ D, ¯ D ∗ ) ↔ ( c ¯ c ) + ( u ( ud ))( p ) (40)To proceed one needs the values of β i , i = 1 , , , A, B, C and a in (30). UsingAppendix 1 one finds the values of ω, Ω in (27), (28) Ω = 1509 MeV, ω = 507MeV. From Appendix 3 one finds the values of β i : β i ( D, D ∗ ) ∼ = β (Σ) = 0 .
48 GeV , β ( p ) ≈ .
26 GeV , (41)and β ( J/ψ ) = 0 . p we have used the principle ofreplacement of light diquark by a light antiquark, ( ud ) → ¯ q . Therefore β ( σ ) = β ( c ( ud ) = β ( c ¯ u ) = β ( D ).As a consequence one obtains the values given in (41). Using those we get thenumerical values for a, A, B, C . a = 12 .
76 GeV − , A = 4 .
02 GeV − , B = 0 .
94 GeV − , | C | < .
03 GeV − (42)As a result one can neglect the C p p in (30) and the integrals d p , d p factorize.We turn now to the recoupling coefficients M ω , ¯ y . As it was shown in [37], the effective parameter M ω can be expressed via the wavefunctions of objects, produced by the string breaking, in our case it is heavy-lightmesons with the coefficient β ( D ) ∼ = β ( B ) = 0 .
48 GeV, and from Eq. (35) of [37]one has M ω ∼ = 2 σβ ( D ) ∼ = 0 . . (43)Finally, the coefficient ¯ χ for the transition into (Σ c ¯ D ) and (Σ c ¯ D ∗ ) can be esti-mated as in the Appendix 2 to be equal to 1. VI. NUMERICAL RESULTS AND DISCUSSION
As was discussed in section 3, (31), the differential cross section for the productionof hadrons in channel 1 can be written as dσdE = | F ( E ) f ( E ) | p ( E ) (44)where F ( E ) is the production amplitude of channel 1 particles without final state F S interaction and f is the F S interaction,which we take as an infinite sum oftransitions from channel 1 to channel 2-the Cornell-type mechanism [28–30]. f ( E ) can be written as f ( E ) = 11 − N ( E ) (45)where N ( E ) has the form N ( E ) = λI ( E ) I ( E ) , (46)where I i ( E ) has the form I i ( E ) = Z d p i (2 π ) v i ( p i ) E ′ ( p i ) + E ”( p i ) − E (47)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 inverseof ( p + ν ). To simplify matter we shall consider situation close to nonrelativisticfor the energies in the denominator of (47) and write E ′ ( p ) + E ”( p ) = m + m + p µ , E ′ ( p ) + E ”( p ) = m + m + p µ (48)We have considered above in the paper v i ( p ) as a Gaussian exp( − b i p i with b =2 B, b = 2 A , see (30). We simplify below this expression, writing exp( − b i p ) = b i p ) ≈ b − i p + ν i , where ν i = √ b i . As a result one can write for I i ( E ) in the region E > E i ( th ) = m + m ( i = 1) or m + m ( i = 2) I i ( E ) = µ i πb i ν i − i p µ i ∆ i , ∆ i = E − E i ( th ) . (49)As a result one obtains a simple expression for the amplitude f ( E ) f ( E ) = 11 − λ ′ µ µ ( ν − i √ µ ∆ ( ν − i √ µ ∆ ) = 11 − zt ( E ) t ( E ) . (50)Here z = λµ µ b b (2 π ) and the (50) refers to the region E > E (th) , E (th) otherwiseone should replace − i before the √ terms by (+1) for the roots, where ∆( E ) isnegative.We now consider the channel coupling constant λ ′ , which enters (50). From (30)one obtains λ ′ = M ω χ b b π a ( Q i β i ) (51)The resulting z is larger than unity for the recoupling coefficient χ of the orderof 1, and one can vary z in the interval from one to larger values.We can now consider 3 transitions, partly discussed above: J/ψ + φ → D ∗ s + ¯ D ∗ s E (th) = 4 .
12 GeV , E (th) = 4 .
224 GeV , µ = 0 . , µ = 1 . , ν =0 . , ν = 0 .
87 (all GeV). J/ψ + p → Σ c (2455) + D (1864) E (th) = 4 . , E (th) = 4 . , µ = 0 . , µ = 1 . , ν = 0 . , ν = 0 . ∗ c (2529) + D ∗ (2010) instead of Σ c + D .As a special interesting case we consider below the recent experiment of BESIII [71], e + + e − → K + ( D − s D ∗ + D ∗− s D )Applying here our recoupling mechanism, shown in the Fig. 1, one easily findsthat the second channel obtained from the first channel D − s D ∗ by recoupling isthe channel J/ψ + K ∗− which creates the chain of reactions possibly generatinga peak in the system D − s D ∗ or D ∗− s D , namely. D − s + D ∗ s → J/ψ + K ∗− E (th) = 3 . , E (th) = 3 . , µ = 0 . , µ = 0 . , ν = 0 . , ν = 0 . ν i and the values of µ i are inthe range 0 . − .
06 GeV. Applying the (50) one can find the f in all cases andhence dσdE in the cases assuming z as a positive number typically largeror equal 1. However for the one should use also the sum: f → f + αf ′ ,implying possible superposition of intermediate states in the rescattering series.We proceed now with the cases and insert the values of E i ( th ) , µ i , ν i in(46,50) and fixing the value of z one obtains the form of the recoupling amplitudeshown in the Tables 2-4 below.For the case the resulting values of | f ( E ) | can be seen in the Table 2. TABLE II: The values of the | f ( E ) | near the channel thresholds for the transition E (GeV) 4.0 4.05 4.12 4.17 4.224 4.3 | f ( E ) | One can see in the Table 2 a strong enhancement around E = 4 .
12 GeV with thewidth around 10 MeV which can be associated with the resonance χ c (4140) havingthe mass 4147 MeV and the width Γ = 22 MeV.In a similar way we obtain the results for the J/ψ + p transitions of . In thiscase we consider the first channel only to simplify matter. TABLE III: The values of the | f ( E ) | near the the Σ D threshold E (GeV) 4.04 4.15 4.25 4.315 4.5 4.7 | f ( E ) | One can see in Table 3 a strong peak near E = 4 . z in the region from 3 to infinity, whichcorresponds to our previous estimates λ ′ >>
1. In this way our method can supportthe origin of the pentaquark state P c (4457) as due to the J/ψ + p ↔ Σ c + D ∗ transitions.We come now to the recent interesting discovery of the new state Z cs (3985) [71],where we take for simplicity only the first chain denoted as the above. Similarlyto the previous cases one obtains TABLE IV: The values of the transition probability as a function of energy in the transition E (GeV) 3.96 3.975 3.98 3.985 3.992 4.0 | f ( | ( z = 1) 5.43 3.45 67.56 31.84 12.95 5.54 | F ( | ( z = 1 .
5) 19.9 236.6 3.19 2.58 1.61 2.71
One can see in Table 4 a narrow peak with the summit at E = 3 .
98 GeV for z = 1with the width around 10 MeV, which closely corresponds to the experimental datafrom [71] E = 3 . . , Γ = 12 . z . As seen in the Table 4, for z = 1 . E = 3 . z = 0 . E = 3 .
992 GeV withsomewhat larger width. In this way we can explain the newly discovered resonance Z cs (3985) by the recoupling mechanism in the rescattering series of transitions D − s + D ∗ s → J/ψ + K ∗− . VII. CONCLUSIONS AND AN OUTLOOK
As it was shown above, the new mechanism having the only parameter z is ableto predict and explain the resonances in different systems,meson-meson,meson-baryon,as it was shown above, and possibly in other systems which can transferone into another via the recoupling of the confining strings. The necessary con-ditions for the realization of these transitions and the appearance of a resonanceare connected to the value of the transition coefficient z ,which should be of theorder of unity or larger. Therefore the transition should be strong,i.e. withoutserious restructuring of the hadrons involved,since otherwise the transition will bestrongly suppressed e.g. in the case when not only strings are recoupled,but alsospins,orbital momenta,isospins should be exchanged. In any case the suggestedmechanism provides an alternative to the popular tetra- and pentaquark mecha-nisms, which dominate in the literature. One should stress at this point,that theindependent and objective checks,e.g. the lattice calculations do not give strongsupport for the molecular or tetraquark models and this topic should be studiedmore carefully. As to the recoupling mechanism, it is strongly associated with thethresholds participating in the transitions, and the best situation for its applicationis both thresholds are close by. In all 3 cases considered above the distance betweenthreshold was less than 30 MeV, and in all cases one could see a strong enhancementin 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 coef-ficient z (originally λ ′ ),and 2) the use of a more realistic Gaussian approximationfor the wave functions instead of approximate ν i parametrizations to define f withgood accuracy in the future. The author is grateful to Lu Meng for an importantremark and to A.M.Badalian for discussions and suggestions. This work was donein the frame of the scientific project supported by the Russian Science FoundationGrant No. 16-12-10414. Appendix A1. The center-of-mass coordinates and average quark andantiquark energies in a hadron
Following [36] one can define the c.m. coordinate of a hadron consisting of a quark Q at the point x and an antiquark ¯ q at the point u via average energies Ω and ω of Q and ¯ q correspondingly as R Q ¯ q = Ω x + ω u Ω + ω ,
Ω = h q p Q + m Q i , ω = h q p q + m q i (A1.1)where q p Q + m Q + q p q + m q is the kinetic part of the Q ¯ q Hamiltonian in theso-called spinless Salpeter formalism or an equivalent form in the so-called einbeinformalism. As a result one obtains the following value of ω = Ω for q ¯ q mesons, shown in TableII. TABLE V: Average values of quark and antiquark kinetic energies in different mesonsState
J/ψ ψ (2 S ) ψ (3770) ψ (3 S ) ψ (4 S )Ω, GeV 1.58 1.647 1.640 1.711 1.17State Υ(1 S ) Υ(2 S ) Υ(3 S ) Υ(4 S ) Υ(5 S )Ω, GeV 5.021 5.026 5.056 5.088 5.120Sate D D s B B s ρ Ω, GeV 1.509 1.515 4.827 4.830 0.4 ω , GeV 0.507 0.559 0.587 0.639 0.4*) Note, that the difference in Ω , ω obtained in these approaches is less or around 1%. see a short recent review in the last refs. in [33] Appendix A2. The channel-coupling coefficient ¯ y We discuss here two topics: 1) the problem of the double string decay vertexcontribution to the recoupling coefficient M ω in (24), 2) the construction of therecoupling vertex ¯ y .We start with the topic 1), and following [37] define the relativistic expression forthe string decay vertex as in (22), which has similar form to the model [ ? ] and thestandard P model [ ? ], but without free parameters namely M ( x ) in (22) is M ( x ) = σ ( | x − x Q | + | x − x ¯ Q | ) . (A2.1)As one can see in Fig.5, in our case the structure of the recoupling process can beexplained by the double string breaking, which we can write as a product S = Z d x ¯ ψ ( x ) ¯ M ( x ) d yψ ( x ) ¯ ψ ( y ) ¯ M ( y ) ψ ( y ) (A2.2)and one must take into account, that the energy minimum of the resulting brokenstring occurs when both time moments x , y of string breaking are equal. Indeed,taking the integral in (A2.2) with account of the string action in the exponent ofthe path integral, ∆ S string = Z σ q r xy + ( x − y ) d x + y , which produces a factor on (A2.2) h r xy i√ π , which denoted as M ω in (24). Theresulting double string breaking action can be written as, S ∼ = Z d x ¯ psi ( x ) ψ ( x ) d y ¯ ψ ( y ) ψ ( y ) M ω ( x , y ) d ( x + y )2 (A2.3)using the notation Z d ( x − y ) h ¯ M ( x ) ¯ M ( y ) i = M ω ( x , y ) . In what follows one can estimate h M ω ( x , y ) i ≡ M ω in the same way, as it wasdone in [37], with the result M ω ≈ σβ , where β is the oscillator parameter for the( Q ¯ Q ) meson. We now turn to the point 2) above, the recoupling vertex ¯ y .To define ¯ y we notice that all 4 quarks q, ¯ q, Q, ¯ Q keep their identity and spinpolarization during the whole process of transformations, provided we neglect thespin dependent corrections. This can be also seen in the structure of the recouplingprocess: in (A2.2) one does not see spin dependence, and this means, that the spin projection of each quark or antiquark is kept unchanged during recoupling. As aresult one can write the nonrelativistic spin part of the matrix element V / as V spinmeson = C Lmµ µ χ (1) µ ¯ χ (2) µ C L ′ m ′ µ ′ µ ′ χ (3) µ ′ ¯ χ (4) µ ′ ( C JMν ν χ (5) ν ¯ χ (6) ν C J ′ M ′ ν ′ ν ′ χ (7) ν ′ ¯ χ (8) ν ′ ) (A2.4)where the Klebsch-Gordon coefficient C JMν ν ≡ C ν ν and χ ( i ) µ , ¯ χ ( k ) λ are quark andantiquark spinors.As was told above, due to the spin conservation in recoupling, the matrix element(A2.4) should be proportional to δ δ δ δ , implying the the recoupling of quarks.As a result one obtains V spin = X µ i ,µ ′ k C Lmµ µ C L ′ m ′ µ ′ µ ′ C JMµ ′ µ C J ′ M ′ µ µ ′ . (A2.5)here L, L ′ , J, J ′ correspond to the spin values of hadrons L + L ′ → J + J ′ and wehave assumed zero orbital momenta for all hadrons. Finally for the final expressionin (24), ( V spin ) should be summed up over all m, m ′ M, M ′ , so that for ¯ y one has h ( ¯ y ) i = X m,m ′ M,M ′ ( V spin ) (A2.6)In a similar way one can find the recoupling coefficient ¯ y for the ensembletransformation p + J/ψ → Σ(Σ ∗ ) + D ( D ∗ ) → p + J/ψ. (A2.7)In this case we write the baryon wave function as ψ B = u ( ud ) , where the lowerindices imply the total spin of the diquark ( ud ). In the simplest approximation onecan approximate the proton as the quark-diquark combination p = u ( ud ) ∼ = u ˜ d ,with the diquark ˜ d kept unchanged d during recoupling. Appendix A3. Oscillator parameters of hadron wave function
The oscillator parameters for the bottomonium, charmonium and
B, D mesonshave been obtained in [32, 33], using the expansion of relativistic wave functions,obtained from the solutions of the relativistic string Hamiltonian [36], in the full setof the oscillator wave functions. As a result one obtainsThe accuracy of the oscillator one-term approximation can be judged by the rel-ative value of the sum od squared coefficients of four higher term of expansion as TABLE VI: The Gaussian parameters β of different mesonsState Υ(1 S ) Υ(2 S ) Υ(3 S ) Υ(4 S ) Υ(5 S ) β , GeV 1.27 0.88 0.76 0.64 0.6State J/ψ ψ (2 S ) ψ (3 S ) ψ (4 S ) ψ (5 S ) β , GeV 0.7 0.53 0.48 0.43 0.41Sate D B ρβ , GeV 0.48 0.49 0.26 compared to the square of the main term. This amounts to the accuracy of the orderor less than 10% for lowest states of charmonia and bottomonia and few percent for
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