Propagators of resonances and rescatterings of the decay products
A.V. Anisovich, V.V. Anisovich, M.A. Matveev, A.V. Sarantsev, A.N. Semenova, J. Nyiri
aa r X i v : . [ h e p - ph ] A ug Propagators of resonances and rescatterings of the decayproducts
A.V. Anisovich + , V.V. Anisovich + , M.A. Matveev + , Nyiri ∗ ,A.V. Sarantsev + , A.N. Semenova + ,October 2, 2018 + National Research Centre ”Kurchatov Institute”: Petersburg Nuclear Physics Institute,Gatchina, 188300, Russia ∗ Institute for Particle and Nuclear Physics, Wigner RCP, Budapest 1121, Hungary
Abstract
Hadronic resonance propagators which take into account the analytical properties ofdecay processes are built in terms of the dispersion relation technique. Such propagatorscan describe multi-component systems, for example, those when quark degrees of freedomcreate a resonance state, and decay products correct the corresponding pole by addinghadronic deuteron-like components. Meson and baryon states are considered, examplesof particles with different spins are presented.
Keywords: Quark model; resonance; exotic states.PACS numbers: 12.40.Yx, 12.39.-x, 14.40.Lb
Nowadays we face a pressing request for studying multi-component systems, in particular, thosewith concurrent parts of quark and hadron degrees of freedom. Recent experimental evidencesfor exotic states (see Refs. [1–3] and references therein) definitely indicate the important role ofboth short-distance physics (predominantly quark-gluon one) and long-distance hadron physicswhere the notion of deuteron-like systems or molecules looks quite appropriate.
4, 5
The activediscussion of the pentaquark topic is in line with this trend.
The two-component structure of resonances can reveal itself in propagators of the res-onances. A corresponding consideration of meson resonances is performed in Ref. [17] fortetraquark systems with hidden charm where meson states for decay processes were taken intoaccount (but with non-relativistic spin wave functions). In this paper we present the relativis-tic consideration of both meson and baryon systems with spins. An important point in thisconsideration is to keep the analytic amplitude with correct singular structure.1 + b + c +... Figure 1: Graphic representation of the Breit-Wigner pole term as an infinite set of transitions resonance state → decay products → resonance state .Let us turn to a standard description of resonances. The Breit-Wigner pole gives usa description of a resonance state in a form of particle propagator, for non-relativistic andrelativistic cases the pole amplitudes read:non − relativistic : G ı G E − E − i Γ2 , (1)relativistic : G ı G M − s − i Γ M .
Here E is the energy of the non-relativistic system, and E is the energy of the resonance level;Γ is the width of the resonance and G ı , G are couplings with initial and final states. Forthe relativistic case the total energy √ s includes the mass of the system and M refers to theresonance mass.The energy independent width corresponds to a rough approximation, for the study of the πp scattering in the ∆(1240) region (hadrons πp are in the P -wave) Gell-Mann and Watson suggested to use the energy dependent width:Γ → γ k πp R k πp , (2)where k πp is the relative momentum of πp in the c.m. system. Actually the width in theform Eq. (2) takes into account the threshold singularity ( k L +1 where L = 1 is the orbitalmomentum) inherent to transitions ∆ → πp → ∆. The threshold singularity appears when weconsider a set of diagrams related to decay processes such as shown in Fig. 1. But Eq. (2)contains also singularities which are absent in the scattering amplitude.First, there are those related to k πp = s [ s − ( m p + m π ) ][ s − ( m p − m π ) ]4 s , (3)namely, the square root singularities at s = 0 and s = ( m p − m π ) ; the form factor 1 / [1 + R k πp ]has a pole singularity which gives zeros in the related amplitudes. For the precise use ofamplitudes with the resonance propagators one needs to take into account the contributions ofdecay processes without false singularities, i.e. to use the corresponding loop diagrams.The paper is assembled as follows. In Section 2 spinless hadron resonances are given andmulti-channel cases are considered. Sections 3 and 4 are devoted to particles with spins, cor-respondingly, to meson and baryon resonances. In Section 5 we investigate the problem of2ormation of the deuteron-like components in states belonging originally to quark-gluon ones.In Appendices A, B, C elements of technique for working with loop diagrams of spin particlesare given. For the inclusion of the loop diagrams into the resonance propagator the D-function techniqueis appropriate, we use it here. First, we consider scalar mesons, after that we generalize theconsideration to cases of spin particles.
The set of diagrams of Fig. 1 reads:1 − s + m + 1 − s + m B ( s ) 1 − s + m + 1 − s + m B ( s ) 1 − s + m B ( s ) 1 − s + m + ... = 1 − s + m − B ( s ) , (4)where B ( s ) is the contribution of the loop diagram related to the resonance decay (in Fig. 1it is supposed that we deal with a two-particle decay). If a resonance state decays into severalchannels, one should replace: B ( s ) → n X ℓ =1 B ( ℓ ) ( s ) (5)where n is the number of open channels. In the standard Breit − Wigner approach the s -independent loop diagrams are used: M = m − n X ℓ =1 ReB ( ℓ ) ( M ) , M Γ = n X ℓ =1 ImB ( ℓ ) ( M ) . (6)Following the Gell-Mann − Watson prescription one takes into account the s -dependent imag-inary part of the loop diagrams: ImB ( ℓ ) ( s ) = ρ ℓ ( s ) g ℓ ( s ) , ρ ℓ ( s ) = k L ℓ +1 ℓ π √ s , (7)where ρ ℓ ( s ) is the phase space for loop-diagram particles, g ℓ ( s ) is the vertex for the transition resonance state → decay particles of the ℓ -state , and L ℓ is the orbital momentum of particlesin the loop diagram. But, as it was discussed above, the imaginary part alone contains falsesingularities. For reproducing the analytical amplitude correctly, one needs to take into accountthe real part of B ( ℓ ) ( s ) as well. 3 .2 D-function and loop diagrams with L = 0 for the one-pole am-plitude Let us consider a loop diagram above the threshold, at s > ( M a + M b ) . The equation forone-pole and one-channel D-function reads: D = d + D B d , d = 1 m − s , B = ∞ Z ( M a + M b ) ds ′ π G ρ ( s ′ ) s ′ − s − i . (8)Here m is a bare mass of this state, the factor B goes from the loop diagram formed by hadrons( a, b ), and M a , M b are masses of the loop mesons. The phase space factor for the S -wave state( L = 0) is: ρ ( s ) = q [ s − ( M a + M b ) ][ s − ( M a − M b ) ]16 πs . (9)The convergency of the integral for B ( s ) can be organized either due to introducing a s -dependence of the vertex G → G ( s ′ ) or by switching the subtraction procedure: B ( s ) = ∞ Z ( M a + M b ) ds ′ π · G ρ ( s ′ ) s ′ − s − i → b + ∞ Z ( M a + M b ) ds ′ π [ s − ( M a + M b ) ] · G ρ ( s ′ )( s ′ − ( M a + M b ) )( s ′ − s − i . (10)Imposing G = 1 we write for positive s , s > ( M a + M b ) : B ( s ) = b + β ( M a + M b ) − ss ( M a + M b ) + q [ s − ( M a + M b ) ][ s − ( M a − M b ) ]16 πs × (cid:20) π ln q s − ( M a − M b ) − q s − ( M a + M b ) q s − ( M a − M b ) + q s − ( M a + M b ) + i (cid:21) ,β = − M a − M b π ln M a M b . (11)The point s = ( M a + M b ) is singular. For s < ( M a + M b ) we write q s − ( M a + M b ) → i q ( M a + M b ) − s , the points s = ( M a − M b ) and s = 0 are not singular, the pole singularityat s = 0 is cancelled due to the term with β .The subtraction constant b regulates a value of the meson component near the threshold,i.e. the fraction of the deuteron-like system. The zero value of the deuteron-like fraction isrealized with B ( s ) (cid:12)(cid:12)(cid:12) s =( M a + M b ) = 0, namely at: b = 0.We have eliminated the pole singularity in the loop diagram introducing the cancellationterm βs . Let us remark, however, that the pole singularity in the loop diagram does not violatethe analytical structure of the total amplitude because the poles in the loop diagram do notlead to new singularities but to zeros of the amplitude (see Appendix A).4 .3 One-pole propagator with non-zero orbital momenta of mesons, L = 0 The loop diagram expression of Eq. (11) gives a possibility to write down analogous terms withnon-zero orbital momenta, L ℓ = 0, and taking into account form factor G ℓ ( s ): B L ℓ ℓ ( s ) = k L ℓ ℓ G ℓ ( s ) B ℓ ( s ) G ℓ ( s ) . (12)Factor G ℓ ( s ) is to be chosen in a form without the violation of the analytical structure of theamplitude, for example, one can use the simple exponential form G ℓ ( s ) = G exp( − R ℓ s ). Theexponential form guarantees the convergence of the loop diagrams. One can use the inversepolynomial function as well: G ℓ ( s ) ∼ /P n ( s ) with P n ( s ) = n P ν =0 a ν s ν because zeros of the P n ( s )are not singilar points of the amplitude. The two-pole D -matrix functions can be written as solutions of the following equations: D = d + D B d + D B d ,D = D B d + D B d ,D = D B d + D B d ,D = d + D B d + D B d , (13)that results in the explicit form D = d B d (1 − B d )(1 − B d ) − d B d B , (14) D = d (1 − B d )(1 − B d )(1 − B d ) − d B d B . Here B if = B fi , zeros of the denominator determine positions of the poles. Expressions for D and D are given by the replacement of indices 1 ⇀↽
2. We have D = D and a commondenominator for all D -functions.If B is small we have two separate poles in the region of studies similar to that discussedin the one-pole case. Non-zero B means a mixture of the input pole states and the change oftheir masses and widths. At large B if additional poles can appear, the additional poles meanthe appearance of new two-meson states created by mesons of the loop diagrams. The equation for the D -matrix can be written as follows:ˆ D ( s ) = ˆ d ( s ) + ˆ D ( s ) ˆ B ( s ) ˆ d ( s ) , (15)that gives: ˆ D ( s ) = ˆ d ( s ) 1 I − ˆ B ( s ) ˆ d ( s ) (16)5here ˆ D ( s ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ( s ) D ( s ) D ( s ) · · D ( s ) D ( s ) D ( s ) · · D ( s ) D ( s ) D ( s ) · ·· · · · ·· · · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˆ d ( s ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ( s ) 0 0 · · d ( s ) 0 · · d ( s ) · ·· · · · ·· · · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˆ B ( s ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B ( s ) B ( s ) B ( s ) · · B ( s ) B ( s ) B ( s ) · · B ( s ) B ( s ) B ( s ) · ·· · · · ·· · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (17)and d i ( s ) = 1 m i − s , (18) B if ( s ) = X ℓ k L ℓ ℓ G iℓ ( s ) B ℓ ( s ) G ℓf ( s ) . Recall that for the one-pole case B ℓ ( s ) ≡ B { ab } ( s ) ≡ B ( s ) is given by Eq. (11). D -function for mesons with spin We consider here meson resonances with spin. First, as elucidation examples, cases with scalar(S), pseudoscalar (P), vector (V), and tensor (T) particles are considered, after that the D -functions for particles with higher spins are presented. − → [1 − ( k a ) + 0 + ( k b )] S − wave → − To calculate the propagator we should calculate the imaginary part of the loop diagram andrestore the real part using Eq. (11). In this procedure the S -wave terms for the transitions V ( in ) → [ V ( a ) + S ( b ) ] S − wave → V ( fin ) (see Fig. 1) are written as follows: g ⊥ Pαβ m − s + g ⊥ Pαα ′ m − s G ( ab ) ( s ) Z d k d k i (2 π ) × δ ( P − k a − k b ) g ⊥ k a α ′ β ′ ( M a − k a − i M b − k b − i G ( ab ) ( s ) · g ⊥ Pβ ′ β m − s + ... = g ⊥ Pαβ m − s h G ( ab ) ( s ) S V SV ( s ) B ( s ) G ( ab ) ( s ) m − s + ... i = g ⊥ Pαβ m − s − G ( ab ) ( s ) S V SV ( s ) B ( s ) G ( ab ) ( s ) . (19)Here g ⊥ Pαβ = g αβ − P α P β P and3 S V SV ( s ) = g ⊥ Pαα ′ g ⊥ k a α ′ β ′ g ⊥ Pβ ′ α = [2 + ( k a P ) s M a ] , (20)6ith vectors k , k being the mass-on-shell values ( k a = M a , k b = M b ) that result:2( k a P ) = s + M a − M b . (21)The width is determined by the imaginary part of the loop diagram, and S V SV ( s ) is a meromor-phic function. − → [1 − ( k a ) + 0 − ( k b )] P − wave → − For the propagator V ( in ) → [ V ( a ) + π ( b ) ] P − wave → V ( fin ) we write: g ⊥ Pαβ m − s + g ⊥ Pαα ′ m − s G ( ab ) ( s ) Z d k a i (2 π ) × ǫ α ′ γ ′ k a P g ⊥ k a γ ′ δ ′ ǫ δ ′ β ′ P k a ( M a − k a − i M b − k b − i G ( ab ) ( s ) g ⊥ Pβ ′ β m − s + ... = g ⊥ Pαβ m − s h G ( ab ) ( s ) S V πV ( s ) B ( s ) G ( ab ) ( s ) m − s + ... i = g ⊥ Pαβ m − s − G ( ab ) ( s ) S V πV ( s ) B ( s ) G ( ab ) ( s ) , (22)with spin factor S V πV ( s ) determined by mass-on-shell mesons, k a = M a , k b = M b :3 S V πV ( s ) = g ⊥ Pαα ′ ǫ α ′ γ ′ k a P g ⊥ k a γ ′ δ ′ ǫ δ ′ β ′ P k a g ⊥ Pβ ′ α = 2 sM a M b h ( k a P ) s − M a i . (23)Recall that the loop diagram factor B ( s ) is given in Eq. (11). S -wave transition of vector-axial state + → [1 − ( k a ) + 1 − ( k b )] S − wave → + For the transition A ( in ) → [ V ( a ) + V ( b ) ] S − wave → A ( fin ) the second diagram of the set of Fig. 1reads: g ⊥ Pαα ′ m − s iǫ α ′ γ ′ γ ′′ P · G ( ab ) ( s ) Z d k a d k b i (2 π ) δ ( P − k a − k b ) (24) × g ⊥ k a γ ′ δ ′ g ⊥ k b γ ′′ δ ′′ ( M a − k a − i M b − k b − i G ( ab ) ( s ) · ( − i ) ǫ δ ′ δ ′′ β ′ P g ⊥ Pβ ′ β m − s with the notation ǫ δ ′ δ ′′ β ′ P = ǫ δ ′ δ ′′ β ′ β ′′ P β ′′ and g ⊥ Pαβ = g αβ − P α P β P .The spin factor of the second term is equal to: g ⊥ Pαα ′ ǫ α ′ γ ′ γ ′′ P · g ⊥ k a γ ′ δ ′ g ⊥ k b γ ′′ δ ′′ · ǫ δ ′ δ ′′ β ′ P · g ⊥ Pβ ′ β = g ⊥ Pαβ S V a V b A ( s ) . (25)Let us remind that here we mean M a = k a an M b = k b . The resonance propagator iswritten as follows: J = 1 : g ⊥ Pαβ m − s − G ( ab ) ( s ) S V ( a ) V ( b ) A ( s ) B ( s ) G ( ab ) ( s ) . (26)7 .4 The S -wave transitions S → V ( a ) + V ( b ) → S ,and T → V ( a ) + V ( b ) → T The propagator for resonance with J = 0 is: J = 0 : 1 m − s − G ( ab ) ( s ) S V ( a ) V ( b ) S ( s ) B ( s ) G ( ab ) ( s ) , (27) S V ( a ) V ( b ) S ( s ) = 13 G ( s )Γ γ ′ γ ′′ ( ⊥ P ) · O δ ′ γ ′ ( ⊥ k a ) O γ ′′ δ ′′ ( ⊥ k b ) · Γ δ ′ δ ′′ ( ⊥ P ) , Γ δ ′ δ ′′ ( ⊥ P ) = O δ ′′ δ ′ ( ⊥ P ) . Here we introduce the vertex function Γ γ ′ γ ′′ ( ⊥ P ) and denote O γ ′ γ ′′ ( ⊥ P ) = g ⊥ Pγ ′′ γ ′ , see AppendixB for details.We write for J = 2: J = 2 : O α α β β ( ⊥ P ) m − s − G ( ab ) ( s ) S V ( a ) V ( b ) T ( s ) B ( s ) G ( ab ) ( s ) , (28) S ( s, k , k ) = 15 G ( s ) O γ ′ γ ′′ α α ( ⊥ P ) · O δ ′ γ ′ ( ⊥ k ) O δ ′′ γ ′′ ( ⊥ k ) · O α α δ ′ δ ′′ ( ⊥ P ) . The operator for the tensor state, O γ ′ γ ′′ α α ( ⊥ P ), is given in Appendix B. Propagators for baryon resonances can be constructed in a way analogous to that for mesonsbut with some complication, namely: determining one-channel rescattering we face two basicloop functions, B ( s ) and e B ( s ).First, we present several examples of propagators for spin-1/2 resonances. Then the caseswith larger spins ( J > /
2) are discussed. The technique used here for fermions with spins
J > / / state and its decay with the emission of a scalar meson We consider here transitions of the type N ∗ (
12 + ) → [ S (0 + ) + N (
12 + )] → N ∗ (
12 + ).The first two terms of the series shown in Fig. 1 are written as:ˆ P + √ sm − s + ˆ P + √ sm − s · G ( s ) B N ∗ ( s ) G ( s ) · ˆ P + √ sm − s , (29)where the loop function B SNN ∗ ( s ) = B N ∗ ( s ) · √ s has the following form: B SNN ∗ ( s ) = Z d ki (2 π ) ˆ k + M N ( k − M N − i P − k ) − M S − i · ( ˆ P + √ s )8 Z d ki (2 π ) kP ) P ˆ P + M N ( k − M N − i P − k ) − M S − i · ( ˆ P + √ s )= Z d ki (2 π ) kP ) s √ s + M N ( k − M N − i P − k ) − M S − i · √ s . (30)We use k = k ⊥ + ( kP ) P P and ( ˆ P + √ s )( A ˆ P + B )( ˆ P + √ s ) = ( ˆ P + √ s )( A √ s + B ) · √ s ; recall thatthe loop diagram hadrons are mass-on-shell in the imaginary part, and 2( kP ) = s + M N − M S .The propagator for the N ∗ (
12 + )-state reads:ˆ P + √ sm − s − G ( s ) B SNN ∗ ( s ) , (31)with loop function B SNN ∗ ( s ): B SNN ∗ ( s ) = 2[( kP ) B ( s ) + M N e B ( s )] . (32)The loop function B ( s ) given in Eq. (11). The new basic term ˜ B ( s ) reads as follows: e B ( s ) = ˜ b + q [ s − ( M S + M N ) ][ s − ( M S − M N ) ]16 πM N √ s (33) × (cid:20) π ln q s [ s − ( M S − M N ) ] − q ( M N − M S ) [ s − ( M S + M N ) ] q s [ s − ( M S − M N ) ] + q ( M N − M S ) [ s − ( M S + M N ) ] + i (cid:21) . Singularities s = 0 and s = ( M S − M N ) are absent in e B ( s ), the only present singularity is thethreshold one s = ( M S + M N ) . In the determination of the e B ( s ) an uncertainty exists whichis related to zeros of the loop functions; this item is discussed in Appendix A, subsection 7.3.At ( M S − M N ) = 0 the loop function has a simple form: e B ( s ) = ˜ b + i q [ s − ( M S + M N ) ]16 πM N . (34) N ∗ (
12 + ) -state with the emission of a pseudoscalarmeson The propagator for the N ∗ (
12 + )-state with the transition N ∗ (
12 + ) → π (0 − ) + N (
12 + ) → N ∗ (
12 + )taken into account can be written as: ˆ P + √ sm − s − G ( s ) B πNN ∗ ( s ) , (35)where B πNN ∗ ( s ) is determined by the loop diagram N ∗ (
12 + ) → π (0 − )+ N (
12 + ) → N ∗ (
12 + ), namely: B πNN ∗ ( s ) = (36)= Z d ki (2 π ) i ˆ k ⊥ γ (ˆ k + M N ) iγ ˆ k ⊥ ( k − M N − i P − k ) − M π − i · ( ˆ P + √ s )9 Z d ki (2 π ) − k ⊥ ( ( kP ) P ˆ P + M N )( k − M N − i P − k ) − M π − i · ( ˆ P + √ s )= Z d ki (2 π ) ( ( kP ) P − M N )( ( kP ) P √ s + M N )( k − M N − i P − k ) − M π − i · √ s . Recall, we use k = k ⊥ + ( kP ) P P and ˆ P → √ s .The hadron rescattering factor is B πNN ∗ ( s ) = 2( ( kP ) P − M N ) h ( kP ) B ( s ) + M N e B ( s ) i , (37)with B ( s ) and e B ( s ) given in Eqs. (11) and (33). ∆(
32 + ) → [ N (
12 + ) + π (0 − )] → ∆(
32 + ) The propagator of the ∆(
32 + )-resonance, taking into account the transition ∆(
32 + ) → N (
12 + ) π (0 − )(see Chapter 5 of ref. [ ] and references therein) reads:( − g ⊥ µν + γ ⊥ µ γ ⊥ ν )( ˆ P + √ s ) m − s − G ( s ) B πN ∆ ( s ) , (38)see Appendix C for details. The factor B πN ∆ ( s ) is determined by the P -wave loop diagram∆(
32 + ) → [ N (
12 + ) π (0 − )] P − wave → ∆(
32 + ), namely:( − g ⊥ µν + 13 γ ⊥ µ γ ⊥ ν ) ( ˆ P + √ s ) B πN ∆ ( s ) (39)= ( − g ⊥ µµ ′ + 13 γ ⊥ µ γ ⊥ µ ′ )( ˆ P + √ s ) Z d ki (2 π ) k ⊥ µ ′ (ˆ k + M N ) k ⊥ µ ′ ( k − M N − i P − k ) − M π − i × ( − g ⊥ ν ′ ν + 13 γ ⊥ ν ′ γ ⊥ ν ) ( ˆ P + √ s )= ( g ⊥ µν − γ ⊥ µ γ ⊥ ν )( ˆ P + √ s ) Z d ki (2 π ) k ⊥ (cid:16) ( kP ) P ˆ P + M N (cid:17) ( k − M N − i P − k ) − M π − i
0) 2 √ s . Therefore the πN rescattering factor can be written as ( k ⊥ = M N − ( P k ) P ): B πN ∆ ( s ) = 2 (cid:16) − M N + ( P k ) s (cid:17)h ( kP ) B ( s ) + M N e B ( s ) i , (40)with B ( s ) and e B ( s ) given in eqs. (11) and (33). Let us consider in a more detailed way the case when the pole singularity is located near thethreshold, m ≃ M a + M b . In this situation the deuteron-like component in the resonant statemanifests itself evidently. 10s an example we consider a case of the one-channel and one-pole amplitude, leading to S -wave decay-products. The scattering amplitude ab → ab reads:1 k e iδ sin δ = G m − s − G ( B ℜ ( s ) + ik ) (41)where B ℜ ( s ) is the real part of the loop diagram. Expanding this amplitude in a series overrelative momentum of mesons, k , one has: G [ m − ( M a + M b ) − G B t ] − k M a + M b ) M a M b (1 + G B ′ t ) − ikG (42)= a a r k − ia k , here B t = B ℜ ( s ) s =( M a + M b ) and B ′ t = (cid:16) dB ℜ ( s ) ds (cid:17) s =( M a + M b ) , whereas a is the scattering lengthand r is the effective radius of the ab -system: a = G m − ( M a + M b ) − G B t , (43) r = − M a + M b ) M a M b ( G − + B ′ t ) . At large negative a the system has a stable component (an analog of the deuteron), at positive a the resonance signal appears only in the continuous spectrum (the system is the analog ofthe singlet state in pp ). A small value of [ m − ( M a + M b ) − G B t ] (a large value of | a | ) canexist independently of details of the long-range hadron-hadron interaction.The large density of the levels in multi-particle systems enlarges the probability to face theeffect of appearance of the deuteron-like components. The hadron resonance topic is a key subject for both experimental studies and theoreticalunderstanding in physics of elementary particles. An important point in this subject is thecorrect description of resonances. Using the language of hadron amplitudes this means a correctrepresentation of the analytical structure of amplitudes. First of all, it concerns the propagatorsof resonances.The experimental study of resonances is connected mainly to the investigation of multi-hadron reactions, the simplest reactions are three-particle ones. The description of Dalitz-plotdata faces problems with the simultaneous presentation of resonances from different channelsand the incorporation of requirements of the unitarity and analyticity into phenomenologicalanalyses.The use of three-body equations leads to implementing the analyticity and the three-bodyunitarity into the amplitude; in the non-relativistic case such an implementation can be per-formed using the Faddeev equation while for the relativistic consideration the dispersionrelation technique looks as the most appropriate one. In this case the resonance propagators11ith included decay components are essential. But in the first attempts to write dispersionrelation equations problems appeared in choosing the way of integration.A correct integration over a three-body intermediate state was performed in Ref. [23], thecorresponding consequences of such an integration are discussed already for a long time. A realistic system of equations for coupled channel amplitudes for proton-antiproton annihila-tion at rest [ p ¯ p ] at rest → πππ, ηηπ, K ¯ Kπ was written in Ref. [29] (see also Ref. [28], Chapter 5).A critical issue in the equations is the dispersion relation presentation of two-meson amplitudesand the corresponding resonances.The multi-component structure of the constructed propagators for resonances allows to fixdeuteron-like states. Examples are presented by states with hidden charm. There are severalcandidates for states with long-distant hadronic components: X (3872) → J/ Ψ ππ (nearbythreshold ¯ DD ∗ ) [30] , X (3900) → J/ Ψ π (nearby threshold ¯ DD ∗ ) [31], X (4020) → J/ Ψ π (nearby threshold ¯ D ∗ D ∗ ) [32]. A popular interpretation of these states is that they are meson-meson molecules ( ¯ DD ∗ and ¯ D ∗ D ∗ ). But it is possible that the states have two components,namely, short-range and long-range ones. That happens when a quark-gluon state (presumablya short-range one) is situated (may be accidentally) in the vicinity of the decay threshold.To conclude: the construction of propagators of composite states with decay loop diagramstaken into account is a relevant subject for both experimental and theoretical studies in hadronphysics. Acknowledgement
We thank D.I. Melikhov for useful comments. The paper was supported by grant RSF 16-12-10267.
Appendix A: Loop diagram analyticity
The convergence of the loop diagram, B ( s ), can be guaranteed by introducing the vertex s -dependence or using the subtraction procedure: + ∞ Z ( M a + M b ) ds ′ π ρ α ′ ( s ′ ) s ′ − s − i → B ( s = s ) + + ∞ Z ( M a + M b ) ds ′ π ρ α ′ ( s ′ ) s ′ − s − i · s − s s ′ − s . (44)Here the subtraction procedure is used. In our studies we put s = ( M a + M b ) .We face two types of the imaginary parts for the loop diagrams: Im B ( s ) = q [ s − ( M a + M b ) ][ s − ( M a − M b ) ]16 πs , (45) Im e B ( s ) = q [ s − ( M S + M N ) ][ s − ( M S − M N ) ]16 πM N √ s . Within these imaginary parts we restore the loop diagrams, see Eqs. (11) and (33).12 eson-meson loop diagram
Let us consider the analytical sructure of the B ( s ) in a more detailed way, keeping s =( M a + M b ) .Below the threshold, at ( M a − M b ) < s < ( M a + M b ) , the loop diagram reads: B ( s ) = b + β ( M a + M b ) − ss ( M a + M b ) + i q [ − s +( M a + M b ) ][ s − ( M a − M b ) ]16 πs × (cid:20) π ln q s − ( M a − M b ) − i q − s + ( M a + M b ) q s − ( M a − M b ) + i q − s +( M a + M b ) + i (cid:21) = b + β ( M a + M b ) − ss ( M a + M b ) + i q [ − s +( M a + M b ) ][ s − ( M a − M b ) ]16 πs × (cid:20) − iπ tan − (cid:18) q − s +( M a + M b ) q s − ( M a − M b ) (cid:19) + i (cid:21) . (46)The last line demonstrates the absence of a singularity in s = ( M a − M b ) . Indeed, in thetop-down approach to this point we have: − iπ tan − (cid:18) q − s + ( M a + M b ) q s − ( M a − M b ) (cid:19) + i = − iπ (cid:18) π − tan − q s − ( M a − M b ) q − s + ( M a + M b ) (cid:19) + i ≃ − iπ (cid:18) π − q s − ( M a − M b ) q − s + ( M a + M b ) (cid:19) + i (47)with the corresponding cancellation of the singular terms in Eq. (48). Meson-baryon loop diagram
The meson-nucleon loop diagram e B ( s ) below the threshold, at ( M S − M N ) < s < ( M S + M N ) ,reads: e B ( s ) = e b + i q [ − s +( M S + M N ) ][ s − ( M S − M N ) ]16 πM N √ s × (cid:20) π ln q s [ s − ( M S − M N ) ] − i | M N − M S | q − s + ( M S + M N ) q s [ s − ( M S − M N ) ] + i | M N − M S | q − s +( M S + M N ) + i (cid:21) = e b + i q [ − s +( M S + M N ) ][ s − ( M S − M N ) ]16 πM N √ s × (cid:20) − iπ tan − (cid:18) | M N − M S | q − s + ( M S + M N ) q s [ s − ( M S − M N ) ] (cid:19) + i (cid:21) . (48)13ear s = ( M S − M N ) we write:tan − (cid:18) | M N − M S | q − s + ( M S + M N ) q s [ s − ( M S − M N ) ] (cid:19) = π − tan − q s [ s − ( M S − M N ) ] | M N − M S | q − s + ( M S + M N ) (cid:19) (49)and the meson-nucleon loop diagram e B ( s ) below the threshold is: e B ( s ) = e b + q [ − s +( M S + M N ) ][ s − ( M S − M N ) ]16 πM N √ s × (cid:20) − π tan − q s [ s − ( M S − M N ) ] | M N − M S | q − s + ( M S + M N ) (cid:21) . (50)It is seen that points s = 0 and s = ( M S − M N ) are non-singular. Moreover, at s = ( M S − M N ) we have e B ( s ) = e b , see Eq. (50), that corresponds to zero of the s -dependent part of the loopdiagram. Ambiguities in the determimation of the loop diagrams are related to zeros of B ( s )and e B ( s ). Ambiguites in the determination of the resonance amplitude
The ambiguities of the resonance amplitude are due to CDD-poles [ ] . The resonance ampli-tude with CDD-poles taken into account is written as follows: B ( s )1 − B ( s ) + P n γ n s − s n (51)A redefinition of the type B ( s ) → B ( s )1 + P n γ n s − s n (52)returns us to the used form of amplitudes. But the redefined B ( s ) differs in numbers and thepositions of zeros. Appendix B: Angular momentum operators for two-mesonsystems
We use angular momentum operators X ( L ) µ ...µ L ( k ⊥ ), Z αµ ...µ L ( k ⊥ ) and the projection operator O µ ...µ L ν ...ν L ( ⊥ P ) (see [
27, 28, 33 ]). Let us recall their definition.The operators are constructed from the relative momenta k ⊥ µ and tensor g ⊥ µν . Both of themare orthogonal to the total momentum of the system: k ⊥ µ = 12 g ⊥ µν ( k − k ) ν = k ν g ⊥ Pνµ = − k ν g ⊥ Pνµ , g ⊥ µν = g µν − P µ P ν s . (53)14he operator for L = 0 is a scalar (we write X (0) ( k ⊥ ) = 1), and the operator for L = 1 is avector, X (1) µ = k ⊥ µ . The operators X ( L ) µ ...µ L for L ≥ X ( L ) µ ...µ L ( k ⊥ ) = k ⊥ α Z αµ ...µ L ( k ⊥ ) ≡ k ⊥ α Z µ ...µ L ,α ( k ⊥ ) ,Z αµ ...µ L ( k ⊥ ) ≡ Z µ ...µ L ,α ( k ⊥ ) = 2 L − L (cid:16) L X i =1 X ( L − µ ...µ i − µ i +1 ...µ L ( k ⊥ ) g ⊥ µ i α −− L − L X i,j =1 i We construct spin-dependent propagators which do not change their spin structure with theinclusion of the loop-diagram interaction. The corresponding spin wave functions are eigenfunc-tions for the interaction. In the framework of this procedure we work with the effective massof the system, and this effective mass depends on the energy, M ( s ). For resonance systems wewrite M ( s ) = s , for detail see [ 27, 28, 34, 35 ]. Baryon spin-1/2 wave function The spin-dependent numerator of the D -function reads: X j =1 , ψ j ( p ) ¯ ψ j ( p ) = ˆ p + M ( s ) , X j =3 , ψ j ( p ) ¯ ψ j ( p ) = − (ˆ p + M ( s )) . (61)where M ( s ) is the effective mass of the resonance system. It means that we work with baryonwave functions ψ ( p ) and ¯ ψ ( p ) = ψ + ( p ) γ which obey the following equations for spin-1/2fermions: (ˆ p − M ( s )) ψ ( p ) = 0 , ¯ ψ ( p )(ˆ p − M ( s )) = 0 , (62)Wave functions are normalised as follows: j, j ′ = 1 , (cid:16) ¯ ψ j ( p ) ψ j ′ ( p ) (cid:17) = 2 M ( s ) δ jj ′ ,j, j ′ = 3 , (cid:16) ¯ ψ j ( p ) ψ j ′ ( p ) (cid:17) = − M ( s ) δ jj ′ . (63)The solution of the equation (62) gives us four wave functions: j = 1 , ψ j ( p ) = q p + M ( s ) ϕ j ( σp ) p + M ( s ) ϕ j ! , ¯ ψ j ( p ) = q p + M ( s ) ϕ + j , − ϕ + j ( σp ) p + M ( s ) ! ,j = 3 , ψ j ( − p ) = i q p + M ( s ) ( σp ) p + M ( s ) χ j χ j ! , ¯ ψ j ( − p ) = − i q p + M ( s ) χ + j ( σp ) p + M ( s ) , − χ + j ! , (64)where ϕ j and χ j are two-component spinors normalised as ϕ + j ϕ j ′ = δ jj ′ and χ + j χ j ′ = δ jj ′ .16olutions with j = 3 , j = 3 , ψ cj ( p ) = C ¯ ψ Tj ( − p ) , C − γ µ C = − γ Tµ . (65)We see that ψ cj ( p ) satisfies the equation:(ˆ p − M ( s )) ψ cj ( p ) = 0 . (66) Spin- wave functions To describe resonance states ∆ and ¯∆, we use the wave functions ψ µ ( p ) and ¯ ψ µ ( p ) = ψ + µ ( p ) γ which satisfy the following constraints:(ˆ p − M ( s )) ψ µ ( p ) = 0 , ¯ ψ µ ( p )(ˆ p − M ( s )) = 0 ,p µ ψ µ ( p ) = 0 , γ µ ψ µ ( p ) = 0 . (67)Here ψ µ ( p ) is a four-component spinor and µ is a four-vector index. Sometimes, to underlinespin variables, we use the notation ψ µ ( p ; j ). Wave function for ∆The equation (67) gives four wave functions for the ∆: j = 1 , ψ µ ( p ; j ) = q p + M ( s ) ϕ µ ⊥ ( j ) ( σp ) p + M ( s ) ϕ µ ⊥ ( j ) ! , ¯ ψ µ ( p ; j ) = q p + M ( s ) ϕ + µ ⊥ ( j ) , − ϕ + µ ⊥ ( j ) ( σp ) p + M ( s ) ! , (68)where the two-component spinors ϕ µ ⊥ ( a ) are determined to be perpendicular to p µ thus keepingfor ∆ four independent spin components µ z = 3 / , / , − / , − / S = 3 / S = 1 / wave functions can be written as follows: X j =1 , ψ µ ( p ; j ) ¯ ψ ν ( p ; j ) = (ˆ p + M ( s )) (cid:18) − g ⊥ µν + 13 γ ⊥ µ γ ⊥ ν (cid:19) = (ˆ p + M ( s )) 23 (cid:18) − g ⊥ µν + 12 σ ⊥ µν (cid:19) , (69)where g ⊥ µν ≡ g ⊥ pµν and γ ⊥ µ = g ⊥ pµµ ′ γ µ ′ . The factor (ˆ p + M ( s )) commutates with ( g ⊥ µν − γ ⊥ µ γ ⊥ ν )in (69) because ˆ pγ ⊥ µ γ ⊥ ν = γ ⊥ µ γ ⊥ ν ˆ p . The matrix σ ⊥ µν is determined in a standard way, σ ⊥ µν = ( γ ⊥ µ γ ⊥ ν − γ ⊥ ν γ ⊥ µ ). Wave function for ¯∆The anti-delta, ¯∆, is determined by the following four wave functions: j = 3 , ψ µ ( − p ; j ) = i q p + M ( s ) ( σp ) p + M ( s ) χ µ ⊥ ( j ) χ µ ⊥ ( j ) ! , ¯ ψ µ ( − p ; j ) = − i q p + M ( s ) χ + µ ⊥ ( j ) ( σp ) p + M ( s ) , − χ + µ ⊥ ( j ) ! . (70)17he completeness conditions for spin- wave functions with j = 3 , X j =3 , ψ µ ( − p ; j ) ¯ ψ ν ( − p ; j ) = − (ˆ p + M ( s )) (cid:18) − g ⊥ µν + 13 γ ⊥ µ γ ⊥ ν (cid:19) = − (ˆ p + M ( s )) 23 (cid:18) − g ⊥ µν + 12 σ ⊥ µν (cid:19) . (71)The equation (70) can be rewritten in the form of (68) using the charge conjugation matrix C which was introduced for spin- particles. We write: j = 3 , ψ cµ ( p ; j ) = C ¯ ψ Tµ ( − p ; j ) . (72)The wave functions ψ cµ ( p ; j ) with j = 3 , p − M ( s )) ψ cµ ( p ; j ) = 0 . (73) Projection operators for resonance states with J > / . The wave function of a resonance state with spin J = ℓ + 1 / 2, momentum p and effective massterm M ( s ) is given by a tensor four-spinor ψ µ ...µ ℓ . It satisfies the constraints(ˆ p − M ( s )) ψ µ ...µ ℓ = 0 , p µ i ψ µ ...µ ℓ = 0 , γ µ i ψ µ ...µ ℓ = 0 , (74)and the symmetry properties ψ µ ...µ i ...µ j ...µ ℓ = ψ µ ...µ j ...µ i ...µ ℓ ,g µ i µ j ψ µ ...µ i ...µ j ...µ ℓ = g ⊥ pµ i µ j ψ µ ...µ i ...µ j ...µ ℓ = 0 . (75)Conditions (74), (75) define the structure of the denominator of the fermion propagator (theprojection operator) which can be written in the following form: F µ ...µ ℓ ν ...ν ℓ ( p ) = ( − ℓ (ˆ p + M ( s ))Φ µ ...µ ℓ ν ...ν ℓ ( ⊥ p ) . (76)The operator Φ µ ...µ ℓ ν ...ν ℓ ( ⊥ p ) describes the tensor structure of the propagator. It is equal to 1for a ( J = 1 / g ⊥ pµν − γ ⊥ µ γ ⊥ ν / J = 3 / γ ⊥ µ = g ⊥ pµν γ ν ).The conditions (5)-(9) are identical for fermion and boson projection operators and thereforethe fermion projection operator can be written as:Φ µ ...µ ℓ ν ...ν ℓ ( ⊥ p ) = O µ ...µ ℓ α ...α ℓ ( ⊥ p ) φ α ...α ℓ β ...β ℓ ( ⊥ p ) O β ...β ℓ ν ...ν ℓ ( ⊥ p ) . (77)The operator φ α ...α ℓ β ...β ℓ ( ⊥ p ) can be expressed in a rather simple form since all symmetry andorthogonality conditions are imposed by O -operators. First, the φ -operator is constructed ofmetric tensors only, which act in the space of ⊥ p and γ ⊥ -matrices. Second, a construction like γ ⊥ α i γ ⊥ α j = g ⊥ α i α j + σ ⊥ α i α j (remind that here σ ⊥ α i α j = ( γ ⊥ α i γ ⊥ α j − γ ⊥ α j γ ⊥ α i ) gives zero if multipliedby an O µ ...µ ℓ α ...α ℓ -operator: the first term is due to the traceless conditions and the second one tosymmetry properties. The only structures which can then be constructed are g ⊥ α i β j and σ ⊥ α i β j .18oreover, taking into account the symmetry properties of the O -operators, one can use anypair of indices from sets α . . . α ℓ and β . . . β ℓ , for example, α i → α and β j → β . Then φ α ...α ℓ β ...β ℓ ( ⊥ p ) = ℓ + 12 ℓ +1 ( g ⊥ α β − ℓℓ +1 σ ⊥ α β ) ℓ Y i =2 g ⊥ α i β i . (78)Since Φ µ ...µ ℓ ν ...ν ℓ ( ⊥ p ) is determined by convolutions of O -operators, see Eq. (77), we can replacein (77) φ α ...α ℓ β ...β ℓ ( ⊥ p ) → φ α ...α ℓ β ...β ℓ ( p ) = ℓ + 12 ℓ +1 ( g α β − ℓℓ +1 σ α β ) ℓ Y i =2 g α i β i . (79)The coefficients in (79) are chosen to satisfy the constraints (74) and the convolution condition:Φ µ ...µ ℓ α ...α ℓ ( p )Φ α ...α ℓ ν ...ν ℓ ( p ) = Φ µ ...µ ℓ ν ...ν ℓ ( p ) . (80) References [1] S. Amato et al., Summary of the 2015 LHCb workshop on multi-body decays of D and B mesons , arXiv:1605.03889 [hep-ex].[2] LHCb, R. Aaij et al., Phys. Rev. Lett. Pentaquarks, PDG (2016).[4] M.B. 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