Meson baryon components in the states of the baryon decuplet

We apply an extension of the Weinberg compositeness condition on partial waves of $L=1$ and resonant states to determine the weight of meson-baryon component in the $\Delta(1232)$ resonance and the other members of the $J^P= \frac{3}{2}^+$ baryon decuplet. We obtain an appreciable weight of $\pi N$ in the $\Delta(1232)$ wave function, of the order of 60 \%, which looks more natural when one recalls that experiments on deep inelastic and Drell Yan give a fraction of $\pi N$ component of 34 \% for the nucleon. We also show that, as we go to higher energies in the members of the decuplet, the weights of meson-baryon component decrease and they already show a dominant part for a genuine, non meson-baryon, component in the wave function. We write a section to interpret the meaning of the Weinberg sum-rule when it is extended to complex energies and another one for the case of an energy dependent potential.


I. INTRODUCTION
The investigation of the structure of the different hadronic states is one of the most important topics in hadron spectroscopy. In order to describe the rich spectrum of excited hadrons quoted in the PDG [1], the traditional concept that mesons and baryons are composed, respectively, by two or three quarks, has been replaced by more complex interpretations, like the ones involving more quarks [2,3].
A remarkable success in describing hadron structure has been obtained by chiral perturbation theory (χP T ) [4,5], an effective field theory in which the building blocks are the ground state mesons and baryons. The low energy processes are well described in this framework, but its limited energy range of convergence makes it unsuitable to deal with higher energies.
Combining unitarity constraints in coupled channels of mesons and baryons with the use of chiral Lagrangians, an extension of the theory to higher energies was made possible. The resulting theory, usually referred to as chiral unitary approach [6][7][8][9][10][11][12][13][14][15][16][17][18], allows to explain many mesons and baryons in terms of the meson-meson and meson-baryon interactions provided by chiral Lagrangians, interpreting them as composite states of hadrons. This kind of resonances are commonly known as "dynamically generated".
An interesting challenge in the study of the hadron spectrum, is understanding whether a resonance can be considered as a composite state of other hadrons or else a "genuine" state. An early attempt to answer this question was made by Weinberg in a time honored work [19], in which it was determined that the deuteron was a composite state of a proton and a neutron. Other works on this issue are [20][21][22]. However, the analysis was made in the case of s-waves and for small binding energies. An extension to larger binding energies, using also coupled channels and in the case of bound states, was done in [23], while in [24] also resonances are considered.
In a recent paper, the work was generalized to higher partial waves [25] and the results obtained were used to justify the commonly accepted idea that the ρ meson is not a ππ composite state but a genuine one. The same method was also successfully used in [26] to evaluate the weight of composite Kπ state in the K * wave function. However, no attempt was done to apply the method to baryonic resonances. We use it here to investigate the nature of the baryons of the J P = 3 2 + decuplet.
The paper proceeds as follows. In Section II we make a brief summary of the formalism. In Section III we address the problem of πN scattering in the ∆(1232) region. In Section IV we extend the test to all the particles of the decuplet while Section V is devoted to discuss and interpret the meaning of the Weinberg sum-rule when extended to complex energies. Finally, we make some conclusions in Section VI.

II. OVERVIEW OF THE FORMALISM
The creation of a resonance from the interaction of many channels at a certain energy, takes place from the collision of two particles in a channel which is open.
The process is described by the set of coupled Schrödinger equations, where H 0 is the free Hamiltonian and µ i is the reduced mass of the system of total mass M i = m 1i +m 2i . The state |Φ 1 is an asymptotic scattering state which is used to create a resonance which will decay into other channels.
Since we shall use this in the discussion later on, it is worth stressing that the wave function is defined up to a global phase, the same for |Ψ and |Φ , as one can see in Eq.(1). However, the standard prescription is to take for Φ the e i k r plane wave function, which then determines the phase of Ψ. We shall come back to the question of phases when we use wave functions in the following.
Following [25], we take as the potential V where Λ is a cutoff in the momentum space and v is a N × N matrix, with N the number of channels. The form of the potential is such that the generic l-wave character of the process is contained in the two factors | p | l and | p | l , and in the Legendre polynomial P l (cos θ), so that v can be considered as a constant matrix. The N × N scattering matrix, such that T Φ = V Ψ, can be written as and the Schrödinger equation leads to the Lippmann-Schwinger equation for T (T = V + V GT ), by means of which one obtains The matrix G in Eq. (5) is the loop function diagonal matrix for the two hadrons in the intermediate state (see Eq. (6)). Note that the definition T Φ = V Ψ makes T independent of the phase convention on the wave function. The derivation in [25] leads to a t matrix which does not contain the factor | p | l , since now the potential v is a constant. Differently from other approaches for p-waves, like the one of [27,28], which factorize on shell | p | l and associate it to the potential v, this factor is now absorbed in a new loop function which is different from the one normally used in the chiral unitary approach [29]. This choice is necessary for the generalization of the sum-rule for the couplings, found in [23] for the case of s-waves, to any partial wave. Indeed, as shown in [25], for a resonance or bound state dynamically generated by the interaction in coupled channels of two hadrons, the following relationship holds (see an alternative derivation in [30] where E R is the position of the complex pole representing the resonance and g i is the coupling to the channel i defined as Note that this definition leads to complex couplings and the sum rule that we derive is obtained in terms of them. In Section V we shall rewrite Eq. (7) for complex energies and discuss the meaning of each term. We anticipate here that each term represents the integral of the wave function squared (not the modulus squared) of each component, but this occurs only in a certain phase convention for the wave function that we shall then discuss. The terms of Eq. (7) are complex, which means that the imaginary parts cancel and then one has and knowing the meaning of these terms, we can consider each one of them as a measure of the relevance or the weight of a channel in the wave function of the state, but not a probability, which for open channels is not a useful concept since it will diverge. Sometimes, our knowledge of all needed coupled channels will be incomplete and we shall only have information on hadron-hadron scattering. There can be a genuine component different to the hadron-hadron one that we study. In order to take into account the weight of this genuine component, Eq. (7) can be rewritten as where Ψ β (p) is the genuine component in the wave function of the state, when it is omitted from the coupled channels. Note that the fixing of a phase in the wave function of one channel will determine the phase of the other wave functions in a coupled set of Lippman-Schwinger equations (see Eqs. (1) and (2)).
The left-hand side of Eq. (10) is the measure of this weight of hadron-hadron component, while its diversion from unity measures the weight of something different in the wave function.
The interpretation of Z as a probability for the non meson-baryon component is rigorous for bound states. For poles in the complex plane we have to reinterpret these numbers, as we have mentioned and will be amply discussed in Section V.

III. πN SCATTERING AND THE ∆(1232) RESONANCE
As already mentioned in the Introduction, the sum-rule of Eq. (10) has been successfully applied to the ρ and K * mesons in [25] and [26], respectively. We use it for the first time to investigate the nature of a baryonic resonance, the ∆(1232), in order to quantify the weight of πN in this state.
We first use a model based on chiral unitary theory, and then, we perform a phenomenological test which makes use only of πN scattering data.

A. The model dependent test
Following the approach of [25,26] we use a potential of the type where √ s R is the bare mass of the ∆ resonance and α and β are two constant factors. Note that we are putting explicitly a CDD pole in v in order to accommodate a possible genuine component of the ∆(1232) in its wave function [31]. In order to account for the p-wave character of the process, the potential v is not dependent on the momenta of the particles. Now, we fit the πN data for the phase shifts using Since the pion is relativistic in the decay of the ∆(1232), we generalize the equations as already done for the case of ρ → ππ in [25]. We take only the positive energy part of the relativistic generalization of the loop function, modified to contain the | q | 2 factor (see Eq. (6) and [25] for more details), with M N the mass of the nucleon, m π the mass of the pion, E N (q) = q 2 + M 2 N and ω(q) = q 2 + m 2 π . The loop function in Eq. (13) is regularized by the cutoff θ(Λ − | q |) of the potential (see Eq. (3)), hence Λ plays the role of q max in the integral of Eq. (13).
To be more in agreement with a propagator which has a denominator linear in the energy, we slightly modify Eq. (11) as where the factor 1/M 4 ∆ is introduced in order to have both parameters, α and β, in units of M eV .
The πN phase shift is given by the formula [32] with p the momentum of the particles in the center of mass reference frame. From the best fit to the πN data we obtain the following values of the four parameters: The results of the fit are shown in Fig. 1. Now we want to apply the sum-rule of Eq. (10) to our case. We need to extrapolate the amplitude to the complex plane and look for the complex pole √ s 0 in the second Riemann sheet. This is done by changing G to G II in Eq. (12), to obtain t II . The function G II is the analytic continuation of the loop function in the second Riemann sheet and is defined as with G I and G II the loop functions in the first and second Riemann sheet, and G I given by Eq. (13).
We are now able to obtain the couplingg ∆ as the residue in the pole of the amplitudẽ and to apply the sum-rule of Eq. (10) to evaluate the πN contribution to the ∆ resonance with Z the weight of something different from a πN state in the ∆. The value of the pole that we get for the best fit is while for the coupling we find From these values we finally obtain which indicates a sizeable weight of πN in the resonance.

B. The phenomenological test
Now we want to evaluate the same quantity using a more phenomenological approach. We repeat the analysis of [25,26] to test the sum-rule by means only of the experimental data.
The ∆ amplitude in a relativistic form is given by where is the three-momentum of the particles in the center of mass reference frame, and The values of M ∆ and Γ on are known from the experiment. Defining √ s = a + ib and making the substitution p → −p in the width term, we obtain the amplitude t ∆ in the second Riemann sheet. Then, we proceed as before to get the pole and the coupling.  The values we obtain for the pole and the coupling, √ s 0 = (1208.00 + i40.91) M eV , are very similar to those obtained with the procedure of the former subsection.
In this case we do not know the size of the cutoff q max needed to regularize the loop function, but the derivative of G II in Eq. (19) is logarithmically divergent in the case of p-waves. Then, using natural values for the cutoff, as done in [25,26], we can establish the stability of the results in a certain range of q max .
The values of 1 − Z for three different values of q max are shown in Table I. They are rather stable and consistent with the result obtained in the previous section.

IV. APPLICATION TO OTHER RESONANCES
Now we extend the study of the hadron-hadron content of resonances to the whole J P = 3 2 + baryons decuplet.
We proceed as in the case of the ∆(1232), applying the phenomenological test of Sec. III B to the other particles of the decuplet, Σ(1385), Ξ(1535) and Ω − .
We first investigate the πΛ and πΣ content of the Σ(1385) wave function. It is known from the PDG [1] that it couples to these two channels with different branching ratios: 87% and 11.7% , respectively. In order to evaluate the couplings of the resonance to the single channel, the branching ratios must be taken into account, modifying Eq. (27) as follows: where BR (i) is the branching ratio to the channel i, with i = πΛ, πΣ and where On the other hand, the case of the Ξ(1535) is completely analogous to the one of the ∆(1232), since, according to the PDG [1] it couples to the πΞ channel with a branching ratio of 100%. Hence, the coupling g Ξ * ,πΞ is simply given by Eq. (27), doing the substitutions  The case of the Ω − is different since this resonance is stable to strong decays. This means that the on shell amplitude Γ on is zero and this prevents us from evaluating the coupling of the resonance to theKΞ channel using Eq. (27). However, from SU (3) symmetry considerations we can relate the g Ω − ,KΞ coupling to g ∆,πN , since their ratios are simply ratios of Clebsch-Gordon coefficients.
We find that The amplitude in relativistic form is again given by Eq. (24) and, in the case of the Σ(1385) and Ξ(1535), it is extrapolated to the second Riemann sheet in order to evaluate the pole and the new couplings. Since, as already said, the Ω − does not decay through strong interaction, the pole of the amplitude is found on the real axis, with no need to go to the second Riemann sheet. It is then possible to apply the sum-rule, evaluating the derivative of the G function in the position of the pole. To do it we use a cutoff of the same order of magnitude of the one found doing the best fit for the ∆(1232), q max 450 M eV . The results obtained for the three resonances are shown in Table II.

V. INTERPRETATION OF THE SUM-RULE FOR RESONANCES
As we could see, we have obtained values of −g 2 dG II d √ s which are complex, and, thus, cannot literally be interpreted as a probability. In this Section we clarify the meaning of the sum-rule in Eq. (9) and of the value of 1 − Z obtained.
Before we give a general formulation of the sum-rule for complex energies based on the results of [23][24][25], let us visualize it in a particular case with two channels, one of them closed and the other one open. Let us also assume, for simplicity, that the interaction in the closed channel is strong and attractive and let us neglect the diagonal interaction in the open channel (the results are the same without this restriction, only the formulation is a little longer). Thus, we have a potential like in Eq.
We shall also assume for simplicity that |v 12 | |v 11 |. The t matrix is given by Eq. (5), and we find Let us now assume that we have a pole in the bound region of channel 1 and open region of channel 2. Then, the denominator of t in Eq. (34) will be zero but G 2 is complex in the first Riemann sheet with in the non-relativistic formulation, and in the relativistic one of Section III, with Let us assume that the attractive v 11 interaction is strong enough to produce a bound state in channel 1 with energy E 1 , when only this channel is considered. Then, we would have The addition of the interaction v 12 will change this energy and Eq. (35) can be rewritten, taking Eq. (38) into account, as (assume v ij independent of energy) where E R will be the new energy of the system.
Since v 11 < 0 and ∂G 1 ∂E < 0 in the bound region The complex value of G 2 , see Eqs. (36) and (37), was obtained for an energy E + i . We gradually continue along the complex plane making the i finite, i Γ 2 , and Eq. (40) gives which is impossible to fulfill in the first Riemann sheet since G 1 < 0, α > 0 and ImG I 2 , given by Eqs. (36)-(37), is negative. This gives us a perspective of why one has to go to the second Riemann sheet, where k 2 → −k 2 in G 2 , in which case one finds a solution, with Next, let us calculate the couplings g i , where g i g j is the residue of the t ij matrix element at the pole. Applying l'Hôpital rule, we have Let us now see that the sum-rule of Eq. (9) is exactly fulfilled, since we have However, this occurs only at the complex poleẼ R + i Γ 2 using G II 2 , since we have made use of the fact that the denominator in g 2 1 and g 2 2 of Eqs. (44) vanishes for E =Ẽ R + i Γ 2 to apply l'Hôpital rule, which only occurs in the second Riemann sheet.
Note that the sum-rule has appeared with the definition of the couplings of Eq. (8). The explicit form obtained for the couplings in Eqs. (44) shows clearly that they are complex, since both G 1 and G 2 are now complex. Now that we have obtained the couplings, let us rewrite Γ of Eq. (43), derived assuming |v 12 | |v 11 | and neglecting again v 12 versus v 11 , as from which follows where we have used the relativistic formula for ImG 2 of Eq. (37) and Eq. (17). As we can see, we reproduce the formula for the width given by Eq. (27). Now we want to interpret the meaning of the sum-rule. Eq. (45) is a generalization to complex energies of the sum-rule obtained in Eq. (119) of [23] and Eq. (101) of [25] for real energies. There it was interpreted as a consequence of the sum of probabilities of each channel to be unity. For complex values of the energies this interpretation is not possible and this is related to the fact that the eigenstates of a complex Hamiltonian are not generally orthogonal 1 .
Formally the problem is solved using, in this case, a biorthogonal basis. Indeed, let λ n be a complex eigenvalue of H and |λ n the corresponding eigenvector. It satisfies det(H − λ n I) = 0 . Then which means that λ * n is an eigenstate of H † . Let now |λ n be the eigenvector of H † associated to λ * n . The eigenvectors |λ n and |λ n are not equal, but we can see that λ n |H|λ m = λ m λ n |λ m = λ n λ n |λ m , where to get the last term we have applied H as H † to the bra state. Thus which means that |λ m and |λ n are orthogonal for n = m. For the case of n = m, λ n |λ n = 0 and we can choose a normalization and a phase for |λ n and |λ n such that λ n |λ n = 1.
The resolution of the identity is now given by n |λ n λ n |. Furthermore, if we have a symmetric but not hermitian Hamiltonian, as it is our case, then it is trivial to see that |λ n = |λ * n for its wave function. Then, the relationship used to derive the sum-rule in [23,25], must be substituted by Hence, for complex values, the modulus squared of the wave function has to be substituted by its square. The integral of Eq. (53) depends on the prescription used for the phase of Ψ i . Below we show that with the standard phase convention used in [25], Eq. (53) is fulfilled.
Recalling that the wave function for us is given by (omitting the spherical harmonics) we can write but we saw in Eq. (45) that and, hence, we conclude that with the phase and normalization chosen for the wave function in Eq. (54). This clarifies the meaning of the sum-rule. It is the demanded extrapolation to complex energies of the sum of probabilities equal unity for real energies. The modulus square of the wave function is substituted by the square of the wave function. Thus we should interpret −g 2 i ∂G II i ∂E as the extrapolation of a probability into the complex plane, but it is not a probability. Yet, once we have interpreted it as the integrated strength of the wave function squared, we still can think of it as a magnitude providing the weight, or relevance of one given channel in the wave function of a state.
As we can see, the integral d 3 p (Ψ(p)) 2 , given in terms of the coupling g i and ∂G II i ∂E , is a finite but complex quantity.
Since the two terms in Eq. (45) will now be complex, the sum of the imaginary parts will vanish and the sum of real parts will be equal to −1. Thus we have and the sum-rule is fulfilled for the real part of the squared of the wave functions. The evaluation of the integral of (Ψ i (p)) 2 is most easily done in momentum space and concretely in terms of G II . Yet, one would like to have a feeling of what happens in terms of wave functions in coordinate space, even if the integration of (Ψ i (r)) 2 in coordinate space requires extra work and is not convenient. We calculate the wave function in coordinate space in Appendix A and we recall only the basic results that we use here for qualitative purposes.
For r → ∞ one obtains for the open channel, in the non relativistic formulation of Section II and in the first Riemann sheet Defining k R = 2µ 2ẼR and k I = 2µ 2ẼR Γ 4Ẽ R , we can write In the second Riemann sheet, we would substitute k by −k and then Hence the wave function in coordinate space in the second Riemann sheet would even blow up, such that a probability would be infinite. This is actually also the case even if k I = 0. Thus the concept of probability is not useful once we have open channels. Yet, and it has an oscillatory behavior that makes the integral for large values of r vanish in the sense of a distribution, like d 3 r e i p r for p = 0. Of course, the finiteness of the integral is better seen integrating in the space of momenta, as we have seen.

VI. DISCUSSION AND CONCLUSIONS
We have applied the generalized compositeness condition to the decuplet of the ∆(1232) to see the weight of meson-baryon cloud and genuine (presumably three quark) components. It is interesting to see that we find the pole position for the ∆(1232), Eq. (20), in very good agreement with the PDG [1] values.
We clarified here the meaning of the extension of the Weinberg sum-rule for the case of resonances and found that −g 2 ∂G II ∂E measures d 3 p p |Ψ 2 and not d 3 p | p |Ψ | 2 . We found that the integral of the real part of the square of the wave function is the natural quantity to provide a measure of the relevance of an open channel in the wave function, since the integral of the modulus squared diverges, even more in the second Riemann sheet. On the other hand, d 3 p p |Ψ 2 is finite and the sum of these quantities for the different coupled channels is unity, within a certain phase convention, as shown by the generalization of the Weinberg sum-rule.
As to the weight of the πN component in the ∆(1232) wave function, we find values which are relatively high, of the order of 60 %. This number could sound a bit large when one thinks of the ∆(1232) as just a spin flip on the quark spins of the nucleon. Yet, the result is less surprising when one recalls that from Drell Yan and deep inelastic scattering one induces a probability of about 34 % for the πN component in the nucleon [34,35]. When one realizes this, then it also looks less surprising that, unlike the case of the ρ, where the analysis in terms of just the ππ component requires large counterterms beyond the lowest order contribution from the chiral Lagrangians [10,27], in the case of the πN scattering in the ∆(1232) region a description was possible with moderate size of the counterterms [36,37].
We extended the compositeness test to the other members of the decuplet and found a decreasing size of the meson-baryon components when we go to the Σ(1385) and Ξ(1535), indicating that the higher energy members of the decuplet are better represented by a genuine (in principle three quark) component. For the Σ(1385) and Ξ(1535) there are also bound components ofKN andKΛ,KΣ, respectively, which we estimate small compared to the open ones in the limited space allowed due to the decay into the open components. In the case of the Ω − , where only the bound componentKΞ is present, we estimate the weight of the meson-baryon component to be small, of the order of 25 %.
The large pion nucleon cloud in the ∆(1232) indicates that realistic calculations of its properties should take this cloud into account. Even before the present test was done to estimate the weight of πN component in the ∆(1232) wave function, the importance of the meson cloud has been often advocated and one example of it can be seen in the early works on the cloudy bag model [38] or chiral quark model [39]. The work presented here offers a new perspective on this interesting subject and the possibility to become more quantitative than in early works.
We have also taken advantage to find an interpretation of the extension of the Weinberg sum-rule for complex values of the energy. We found that the concept of probability is then changed to the squared of the wave function, within a certain phase convention, which, upon integration, leads to finite values that we present as a measure of the weight of a channel in the wave function, while the modulus squared of the wave function is divergent for open channels.

Appendix A: Wave functions in coordinate space
The wave function in momentum space is given by (let us take also a spherical harmonic Y 10 (p) for simplicity) Ψ( p ) = g θ(Λ − p) p E − p 2 /2µ Y 10 (p) ≡Ψ( p )Y 10 (p) .
(A2) In the second Riemann sheet we change √ 2µE to − √ 2µE. As we can see, for large values of r the integrals over the half circle in Fig. 2 are strongly suppressed by the factor e −Λr sin θ (θ ∈ [0, π]), which makes these integrals vanish when r → ∞.
Then, the dominant term for r → ∞ is given bȳ which has been used in the discussion in Section V.