Lambda(1405) as a K-bar N Feshbach resonance in the Skyrme model

We describe the Lambda(1405) hyperon as a Feshbach resonance of a bar-KN quasi-bound state coupled by a decaying channel of pi Sigma in the Skyrme model. A weakly bound bar-KN state is generated in the laboratory frame, while the Sigma hyperon as a strongly bound state of bar-KN in the intrinsic frame. We obtain a coupling of bar-KN and pi Sigma channels by computing a baryon matrix element of the axial current. This coupling enables the decay of the bar-KN bound state to pi-Sigma . It is shown that the Skyrme model supports the Lambda(1405) as a narrow Feshbach resonance.


I. INTRODUCTION
The negative parity state of the hyperon of the lowest mass, Λ(1405), has brought many discussions over half century, because its properties are not easily explained by the standard quark model [1]. For example, its excitation energy of about 300 MeV above the ground state Λ hyperon with mass 1116 MeV is considerably smaller than the other light flavored baryons of typical excitation energy about 600 MeV, i.e., N (1535) − N (940) ∼ 600 MeV. In fact, before the quark model becomes popular, Dalitz and Tuan analyzed the anti-kaon and nucleon (KN ) scattering data and suggested the existence of a qusi-bound state ofKN corresponding to Λ(1405) [2,3]. To support such a bound state, the interaction betweenK and N must be sufficiently attractive.
Employing the mass of Λ(1405) at the nominal value of 1405 MeV, aKN potential was proposed to reproduce the mass in Refs [4,5] and applied to few-body systems ofK and a few nucleons, resulting in unexpectedly deeply bound states. On the other hand, a chiral model for theKN was developed, which predicted a less attractive interaction that is still sufficient to generate a loosely boundKN state with a mass spectrum of Λ(1405) being consistent with experimental data [6]. In contrast with the former approach, the chiral model does not generate deeply bound strange nuclei. Moreover, a unique feature of the chiral models is that it generates two pole structure for Λ(1405), one is ofKN origin while the other πΣ origin [7,8]. The one of πΣ origin locates at a deep imaginary region on the complex energy plane, resulting in a broad background structure in the spectrum.
These different natures originate from the uncertainties in the basic interaction. The phenomenological interaction is determined by the nominal mass of the Λ(1405). The structure of the interaction such as the ranges and strengths depend much on the data employed. The chiral model that is based on spontaneous breaking of chiral symmetry of QCD still contains parameters for renormalization or subtraction. In both methods, parameters are adjusted to reproduce the existing data for the Λ(1405).
Observing this situation, we have developed an alternative approach in the Skyrme model [10,11]. It is a non-linear field theory with chiral symmetry for mesons, where baryons emerge as solitons [12][13][14][15][16][17][18][19]. The model has been shown to be successful, at least qualitatively, for meson and baryon spectroscopy and their interactions. The advantage of this model is that once the two parameters are fixed from meson properties, the dynamics of baryons are determined without additional parameters. In this manner we expect that we better discuss exotic phenomena such as high density matter with knowing the origin of the dynamics. This is the reason that we employ the Skyrme model in the present study.
In our previous publications [10,11], we have investigated theKN interaction in the Skyrme model using an analogous method to the bound state approach by Callan and Klebanov [20,21]. Their method is formulated following the 1/N c expansion with the collective quantization of solitons and was shown to be successful for the descriptions of the ground state Λ and Σ hyperons. An interesting observation is that theK is strongly bound to the hedgehog soliton in its rest frame (intrinsic frame), and consequentlyK is interpreted as a strange quark with spin 1/2 when quantized. Their method corresponds in many-body physics to the projection after variation, or the strong coupling scheme [22]. In our approach, observing that theK in Λ(1405) is weakly bound to the nucleon, we have proposed an alternative method, that is the method of projection before variation, or the weak coupling scheme. Setting the two parameters at suitable values, the pion decay constant and the Skyrme parameter, it has been shown that theK feels an attractive interaction from the nucleon and is bound with a binding energy of order ten MeV, which is identified with Λ(1405). Another interesting feature is that when theKN interaction is expressed in the form of a local potential, it exhibits an attractive pocket at medium distances supplemented by a repulsion at short distances. These features would influence on the properties of high density matter with kaons.
In this paper, we introduce a coupling ofKN to πΣ to enable theKN bound state to decay, and investigate whether the boundKN state survives as a Feshbach resonance. In terms of the low energy method of chiral symmetry, the one pion emission decay is computed by the baryon matrix element of the axial current. Details of technical issues in performing such a computation is developed.

II. ACTIONS AND ANSATZ
Let us start with the SU(3) Skyrme model action given by [19] The first and second terms are the original Skyrme model actions and the third term is the symmetry breaking term due to finite masses of the pseudo-scalar mesons, In this paper, we treat the pion as a massless particle while the kaon as massive one. The last term in Eq. (1) is the contribution of the chiral anomaly called the Wess-Zumino-Witten action given by [16,17], with N c the number of colors, N c = 3. TheKN system is described by employing an ansatz [20] U where ξ(x) is for the pion field embedded in the upper 2 × 2 components, with F π ∼ 186 MeV the pion decay constant, and U K for the kaon field defined by, The Skyrme model describes the nucleon as solitons of the pion field. The model accommodates a static classical solution with a specific symmetry, that is called the hedgehog solution, where F (r) is a soliton profile function of radius r ≡ |x|, andx = x/|x|. Such a classical solution does not correspond to the physical nucleons with spin and isospin quantum numbers. They are generated in the collective coordinate method, where the variables for spin and isospin rotations of the hedgehog solution are quantized. Therefore, the nucleon is regarded as a rotating hedgehog, Due to the symmetry of the hedgehog solution, rotations in spin and isospin spaces are related leading to the constraint of equal spin (J) and isospin (I) values, J = I.
In the present study for the decay Λ(1405) → πΣ, we need the kaon field that plays dual roles. One is for Λ(1405) where the physicalK of isospin 1/2 is bound to the nucleon, the rotating hedgehog in the laboratory frame. Hence we have proposed an ansatz [10], which is used for the construction of the Λ(1405). The other is for Σ where theK is bound to the hedgehog soliton in its intrinsic rest frame. The total configuration of the hedgehog soliton with a boundK is then rotated simultaneously, where the subscript CK is from Callan-Klebanov [20]. This equation can be written also which explicitly indicates that the hedgehog and kaon are rotating in the same way by the rotation matrix A(t). In terms of the two-component iso-spinor the kaon field is rotated as Intuitively, the two different schemes for theK bound in Λ(1405) and in Σ are understood by comparing the time for the rotating hedgehog to turn around once, ∆t H , and the time of the bound kaon to go around the soliton (nucleon) once, ∆t K . The time ∆t H for the nucleon of spin J = 1/2 is estimated if we know the angular velocity Ω of the rotating hedgehog for the nucleon by ∆t H ∼ 2π/Ω. Using the relation J = IΩ = 1/2 and the moment of inertia value I ∼ 1 fm of the rotating hedgehog, we estimate Ω ∼ 1/2 fm −1 and hence ∆t H ∼ 10 fm. The time ∆t K for Λ(1405) is estimated by using a typical binding energy of theK that is of order ten MeV, while that for Σ is estimated by using a typical binding energy of order hundred MeV. We find the relation ∆t H < ∆t K ∼ a several ten fm (13) for theK of Λ(1405) (theK goes around more slowly than the hedgehog rotates), implying that theK is treated as a particle moving around the rotating hedgehog in the laboratory frame. On the other hand, we find for theK of Σ (theK goes around faster than the hedgehog rotation), implying that theK is treated as a particle moving around the static hedgehog in the intrinsic (rotating) frame.

A. Definitions
The decay of Λ (1405) → πΣ is regarded as a baryon transition accompanied by one pion emission, which is described by the amplitude To the leading order of chiral expansion in powers of small momentum, the interaction Lagrangian with one pion L int is written as The isospin axial current J µ,a 5 with the isospin index a is the one with the one pion pole term subtracted and is computed in the Skyrme model in the present study. We note that it is normalized in accordance with the isospin; for instance, for the effective interaction with the nucleon, J µ,a 5 →ψ N γ µ γ 5 (τ a /2)ψ N . For the transition of Λ (1405) → πΣ we need the isovector axial current in the form, where ψ a Σ and ψ Λ(1405) are the Dirac spinors for Σ and Λ(1405) with a an isospin index for Σ. Here γ 5 is not needed due to the negative parity of Λ(1405). The baryon matrix element computed in the Skyrme model is then identified with the coupling constant for the effective Lagrangian The coupling constant g Λ(1405)πΣ is then defined to be the matrix element the estimation of which is the main purpose of the present paper.

B. The axial current
The axial current is derived from the action Eq. (1) as the Noether's current associated with the axial transformation, where λ = λ a (a = 1, 2, 3, · · · , 8) and θ are the Gell-Mann matrices and SU (3) parameters, respectively. The result is (x = (t, x)) where The current here is regarded as an operator acting on the quantized soliton states written in terms of the collective coordinates of rotations, and on the second-quantized states of the kaon as we will see below.
Substituting the ansatz (4) for (21), and expanding in powers of the kaon field K up to the second order, we find where superscripts (0) and (2) stand for the order of the kaon field. For our purpose, we need the second order term J µ,a,(2) 5 which contains two kaon fields, K and K † . Moreover, in the non-relativistic approximation that we employ for baryons, the time component µ = 0 is dominant. The explicit form of the relevant piece of the first term of (21) is The computation of the second and third terms of (21) is tedious, but possible and is given in Appendix A. As anticipated in the previous section, the dual roles of the kaon fields in (23) are implemented by identifying one of K's in (23) with that for Λ(1405) and the other one for Σ, when computing the matrix element Σ 0 | J 5,a=3 µ=0 |Λ (1405) . Explicitly, we follow the relation K → A(t)K CK for Σ, The presence of collective coordinate A(t) in the first equation is inferred from (12) and is regarded as a coordinate operator. Following the standard method for field quantization, the kaon fields are expanded in terms of a complete set of wavefunctions with the corresponding creation or annihilation operators as their coefficients. The field K † EH is regarded as an annihilation operator for the antikaon for Λ(1405) and is expanded by the wavefunctions in the laboratory frame, where φ's and a's are the wavefunctions and the corresponding annihilation operators, respectively. Here we have shown only the terms of the lowest s-wave for the anti-kaon that are necessary for our purpose, where k(r) is the s-wave radial function of the antikaon bound to the nucleon with E EH being the corresponding energy including its rest mass. The minus sign in the 2nd component for φ †K 0 reflects the proper isospin transformation ofK. For Σ, the kaon is bound to the hedgehog with quantum numbers of the grand spin, the sum of isospin and orbital angular momentum, T = I + L. As discussed in Ref. [20], such a bound kaon is interpreted as a strange quark in p-wave. Therefore, where the p-wave nature is in the combination τ ·x, s(r) the corresponding radial function and E CK the energy. Once again the minus sign in the lower component of the second line of (28) reflects properly the spin transformation rule. The functions k(r) and s(r) are obtained by solving the Klein-Gordon like eigenvalue equations [10,11,20,21]. Their normalization needs to be treated properly to reflect the structure of the Klein-Gordon like equations, as shown in Appendix.

C. Baryon states
The isosinglet state of Λ(1405) is formed by the two isospin 1/2 states of the nucleon and the kaon, The proton (p) and neutron (n) wavefunctions with spin up and down ψ pn,↑↓ (A) are given by the collective coordinate A, .

(30)
The Σ state is given by a combinations of diquark like wavefunctions of spin and isospin 1, and of the strange quark. For neutral spin up Σ, where the vector-isovector diquark wavefunctions are labeled by its spin J 3 and isospin I 3 , ψ d J 3 I 3 , and the relevant ones here are given,

IV. CALCULATION OF THE MATRIX ELEMENT
After establishing the axial current and the wavefunction, we demonstrate how the matrix element (18) is computed. The procedure is rather straightforward, though actual computation is quite long and tedious. Therefore, we will show the outline briefly. Let us consider the transition to the neutral Σ (a = 3). Replace the kaon fields as in (24), and the time derivatives by the eigenenergies of the relevant terms of (26) and (28), we find where we have indicated that the current is a function of x and the collective coordinate A.
For the transition amplitude, we need to take the matrix element of the interaction Lagrangian (16) with the initial Λ(1405) and the final Σπ, with a finite pion momentum q µ = (E π , q), E π = m 2 π + q 2 . Performing necessary trace algebra for the relevant 2 × 2 matrices, we integrate over the space-time d 4 x and collective coordinates dµ(A), where and the relation between the three angles and the SU(2) rotation matrix is given by a 0 = cos θ 1 a 1 = sin θ 1 sin θ 2 cos θ 3 a 2 = sin θ 1 sin θ 2 sin θ 3 a 3 = sin θ 1 cos θ 2 .
The time integral leads to the δ-function for energy conservation. After these manipulations, we arrive at a rather compact expression π 0 (q)Σ 0 | L int |Λ(1405) The presence of the spherical Bessel function j 0 (qr) = sin(qr)/(qr) indicates that the decaying pion is in the s-wave as it should be. So far we have shown the result for the second order derivative term. The computation goes similarly for the Skyrme and WZW terms. The results are summarized as follows, where sin F s(r)k * (r), Here we have introduced the notation c = cos(F/2) and s = sin(F/2).

V. RESULTS AND DISCUSSIONS
In this section, we present and discuss our numerical results for the decay of Λ(1405) → πΣ. The formulae that are derived in the previous sections determine the coupling constant g Λ(1405)πΣ as defined in the effective Lagrangian (18). The decay width is then computed by the formula, The factor 3 is for isospin sum. For kinematic parameters we employ the physical values that are fixed by the experiment as summarized in Table I. Here we take the mass of Λ(1405) slightly higher than the nominal value, that is 1420 MeV, considering the recent discussions of the two pole structure of Λ(1405), and theKN quasi-bound state is considered to locate at around the higher mass region [9].
138 49 1193 1420 166 216 1204 Our main results in this paper are shown in Table II, where various contributions to the coupling constants and the resulting decay widths are given for three sets of the Skyrme model parameters, A, B and C. In Set A, the decay constant F π is taken at an average of the pion and kaon decay constants, while in Set B it is set at the pion decay constant. The Set C is from Ref. [18]. In all cases, the Skyrme parameter e is determined such that the N ∆ mass splitting is reproduced.   As seen from the Table II, the present model predictions of the decay width Γ are small as compared to the experimental data, and scatter in a range from the minimum value to the maximum value that is about three times larger than the minimum value. The experimental data is taken from PDG where they quote the average number 50.5±2.0 MeV [9]. There are, however, discussions about the two pole structure of Λ(1405) having theKN and πΣ origin. TheKN originated one locates relatively higher in mass at around 1420 MeV and has a narrower width, while the πΣ originated one locates lower with a wider width. Our present result is to be compared with the formerKN dominant one, whose width is expected to be around 20 MeV [9]. Thus the corresponding coupling constants are shown in parentheses.
The reason that the model predictions scatter in a rather wide range is that the amplitude is proportional to 1/F π and that the overlap integral in the matrix element is sensitive to the structure of the kaon wavefunctions of Λ(1405) and of Σ. It is not difficult to see that these factors may change the coupling constant by a few times. Then a possible reasons for small values may be explained by the overlap integral; in the present approach the two limits are employed for the construction of the wavefunctions of Λ(1405) and Σ, the weak coupling and strong coupling limits. The matrix elements for the transition amplitudes computed by the integral of the two wavefunctions is therefore suppressed. In realistic situation, the both wavefunctions are between the two limits and therefore the overlap integral would gain some strength. We also consider that the suppression is related to the bound state approach where the kaon is regarded as a heavy meson and is, as well as hyperons, not treated as flavor SU(3) multiplets. Physically, the transition fromKN to πΣ requires an exchange of a (heavy) strange quark fromK to Σ. It is natural to consider that such a heavy particle exchange is suppressed.
Aside from the quantitative aspect, it is worth emphasizing as the main conclusion of the present study that the resulting decay width turns out to be narrow. This enables for thē KN bound state to remain as a Feshbach resonance, seemingly a natural consequence that the Skyrme model supports. where where ρ 1 (r) = − 4 sin 2 (F/2) 3Λ I K · I N 1 + 1 (eF π ) 2 4 r 2 sin 2 F + F 2 − sin 2 (F/2) Λ 1 + 1 (eF π ) 2 5 r 2 sin 2 F + F 2 , (B6) where k m (r) and ω m are the wavefunctions and the corresponding eigenenergies, respectively, and the tilded variables are for the kaon. These normalization conditions Eqs. (B2) and (B6) is obtained in order to satisfied with the canonical quantization condition, [k n (r, t), π m (r , t)] = iδ nm δ (3) (r − r ), where π m (r , t) is the canonical momentum conjugate to k m (r, t).