Electromagnetic mean squared radii of Lambda(1405) in chiral dynamics

The electromagnetic mean squared radii,_E and_M, of Lambda(1405) are calculated in the chiral unitary model. We describe the excited baryons as dynamically generated resonances in the octet meson and octet baryon scattering. We evaluate values of_E and_M for the Lambda(1405) on the resonance pole and obtain their complex values. We also consider Lambda(1405) obtained by neglecting decay channels. For the latter case, we obtain negative and larger absolute electric mean squared radius than that of typical ground state baryons. This implies that Lambda(1405) has structure that K^- is widely spread around p.


Introduction
The structure of the Λ(1405) has been a longstanding issue. The Λ(1405) has been considered as a quasi-bound state of anti-kaon and nucleon (KN ) system [1,2,3], before QCD was established. Recent theoretical investigations have also suggested that the Λ(1405) is well described as a dynamically generated resonant state of meson-baryon scattering with I = 0 and S = −1 based on chiral dynamics in coupled channel approach [4,5,6,7,8,9,10], which is so-called chiral unitary model (ChUM). This model has successfully reproduced cross sections of K − p to various channels and also the mass spectrum of the Λ(1405) resonance below theKN threshold [4,5,6,8,11], giving two states for the Λ(1405) found as poles of the scattering amplitude in the complex energy plane [6,9,12]. Photo-properties of Λ(1405) have been investigated in the chiral unitary approach in Refs. [13,14,15,16].
Most of excited baryons can be described by simple constituent quark models [17]. The Λ(1405) is well known as one of the exceptions, so the success of the hadronic molecule picture is in this sense reasonable. Recent study of the Λ(1405) based on the N c scaling in ChUM [18] indicates the dominance of the non-qqq component in the Λ(1405), and Ref. [10] also suggests that the Λ(1405) is described predominantly by meson-baryon dynamics. The understanding of the structure of the Λ(1405) is also relevant for theKN phenomenology, since binding energy ofKN system in the Λ(1405) plays an important role for the study of the kaonic nuclei [19,20,21,22]. Toward experimental verification, it is desirable to study some quantity which characterizes the molecule structure of Λ(1405) [15].
One expects that the Λ(1405) as meson-baryon quasi-bound molecule with small binding energy has a larger size than typical ground state baryons dominated by a genuine quark state. If this is the case, the form factor of the Λ(1405) falls off more rapidly than that of the nucleon and the production cross section of the Λ(1405) has large energy dependence. In this work, we estimate the electro-magnetic mean squared radii of the Λ(1405) based on the ChUM. Such electromagnetic properties of the Λ(1405) could be obtained in photon-induced production experiments [23].

Λ(1405) in chiral unitary model
In ChUM, the Λ(1405) is described in s-wave meson-baryon scattering amplitudes with the strangeness S = −1 and charge Q = 0 obtained by solving the Bethe-Salpeter (BS) equation, in which the on-shell factorization leads to an algebraic solution [6]. In Eq. (1), V ij is s-wave mesonbaryon interaction kernel and G k is loop integral of the meson-baryon system. They are functions of the center-of-mass energy, √ s, in matrix form of the meson-baryon channel (i, j = K − p,K 0 n, π 0 Λ, π 0 Σ 0 , ηΛ , ηΣ 0 , π + Σ − , π − Σ + , K + Ξ − and K 0 Ξ 0 ). The interaction kernel V ij is given by the leading order chiral Lagrangian [24,25], which is known as Weinberg-Tomozawa term with an averaged meson decay constant f = 1.123f π , f π = 93.0 MeV. This expression is obtained by applying the non-relativistic reduction for baryons. The coefficient C ij is fixed by the SU(3) group structure of the interaction and its explicit value is given in [5]. The loop integral G k ( √ s) is evaluated using the dimensional regularization: where the center-of-mass momentum of the twobody system is given by The subtraction constants a k (µ) in Eq. (3) with the regularization scale µ are free parameters in this model. These constants are phenomenologically fixed so as to reproduce the threshold behavior of the scattering amplitudes [7]: with µ = 630 MeV. In the present model, the Λ(1405) is dynamically generated in the obtained BS scattering amplitude without introducing any explicit pole terms in the interaction kernel. The excited baryon is expressed by a pole of the scattering amplitudes in complex energy plane. The s-wave scattering amplitude can be approximated as, close to the resonance energy, as shown in Fig. 1(a). The real and imaginary parts of the pole position z H express the mass and the half-width of the excited baryon, respectively, and the residues of the pole, g i and g j , represent coupling strengths of the excited baryon to the meson-baryon channels. In the present model, two poles in the BS amplitude are found in energies of the Λ(1405) as (z 1 = 1390 − 66i MeV) and (z 2 = 1426 − 17i MeV) [9]. It has been reported in Ref. [26] that the position of the lower pole z 1 is dependent on details of model parameters, whereas that of the higher pole z 2 shows little dependence.

Form factors of excited baryons
In this section, we discuss the formulation to evaluate the electromagnetic form factors and the mean squared radii of excited baryons described by the BS amplitudes. First of all, let us define the electromagnetic form factors of an excited baryon with spin 1/2, H * , as matrix elements of the electromagnetic current J µ EM in the Breit frame [15]: with the electric and magnetic form factors, G E (Q 2 ) and G M (Q 2 ), the virtual photon momentum q µ , Q 2 = −q 2 and the Pauli matrices σ a (a = 1, 2, 3). The magnetic form factor G M (Q 2 ) is normalized as the nuclear magneton µ N = e/(2M p ) with the proton charge e and mass M p . From these form factors, electromagnetic mean squared radii, r 2 E and r 2 M , are calculated by To extract the matrix elements of the electromagnetic current from the BS amplitudes, we consider the scattering amplitude for the M Bγ * → M ′ B ′ process, T µ γij , which is microscopically calculated by attaching the photon to every place of the constituent meson-baryon components in the BS amplitude [14,15,27]. The matrix elements of the excited baryon can be expressed as residues of the double pole of the M Bγ * → M ′ B ′ amplitude. Close to the pole of the excited baryon, as shown in Fig. 1 where s ′ = P ′µ P ′ µ and s = P µ P µ , with incoming and outgoing momenta of excited baryon, P µ and P ′µ , respectively. Combining Eqs. (6) and (10), the matrix elements can be evaluated as residue of the double pole at s = s ′ = z H as discussed in [15]: (11) Here we calculate residue of √ s ′ = z H by "Res" in the right-hand side. Note that this evaluation is free from the non-resonant background, since the values are calculated just on the pole of the excited baryon.
Thus, we have shown that the electromagnetic mean squared radii of the excited baryon can be obtained, once the scattering amplitude M Bγ * → M ′ B ′ is calculated.

Evaluation of the form factors
We calculate the scattering amplitude of the M Bγ * → M ′ B ′ in the chiral unitary approach, in which the amplitude for the M B → M ′ B ′ is given by multiple scattering of the meson and baryon. Thus, the photon couples to the Λ(1405) through the constituent mesons and baryons. The calculation should be performed in a gauge-invariant way, which ensures the correct normalization of the electric form factor of the excited baryon, G E (Q 2 = 0) = Q H . Following the method of the gauge-invariant calculation for unitarized amplitudes proposed in Ref. [27], we take three relevant diagrams shown in Fig. 2, which have the double pole for the excited baryon. The other diagrams do not contribute to the electromagnetic form factors at the resonance energy √ s = z H . Since the pole position is gauge invariant, these three contributions are enough for the gauge-invariant form factors: Using this amplitude T µ γij with photon couplings to mesons and baryons which we will discuss later, we obtain the gauge-invariant form factors through Eq. (11). A proof of the gauge invariance of the form factors is given in Appendix A by using Ward-Takahashi identity.
The elementary couplings of the photon to the meson and baryon are given by imposing gauge invariance to the chiral effective theory in a consistent way with the description of Λ(1405). In the present model, the Λ(1405) is described by a infinite sum of the loop function G and the s-wave Weinberg-Tomozawa interaction V , with non-relativistic formulation for the baryons. We use the minimal coupling scheme for the photon couplings to the meson and baryon appearing in the BS amplitude. This procedure automatically implements gauge invariance in a consistent way to the original BS amplitude. We also have the anomalous magnetic couplings for the BBγ and γBB ′ M M ′ couplings. These couplings are given by the chiral perturbation theory as done in Ref. [15]. Finally the elementary electric and magnetic couplings, V µ Mi , V µ Bi and Γ µ ij , for M M γ, BBγ and γBB ′ M M ′ , respectively, are obtained by sum of the two contributions.
In the minimal coupling scheme, the photon coupling to the meson, V µ Mi , is given by with the incoming and outgoing meson momenta k µ and k ′µ . The minimal coupling of the photon to the baryon is given by with the normal magnetic moments µ (N) i and the incoming and outgoing baryon momenta p µ and p ′µ .
Here we have performed non-relativistic reduction. These two couplings (13) and (14) are appropriate with the propagators in the loop function (3). The spatial components without the Pauli matrices σ in Eqs. (13) and (14) give no contribute to the Λ(1405) form factors in the Breit frame. For the γBB ′ M M ′ coupling appearing in T µ=0 γ3ij , we use the following vertex which is obtained so that Ward-Takahashi identity is satisfied with Eqs. (2), (13) and (14) in tree-level: with incoming and outgoing meson-baryon momenta, P µ and P ′µ , respectively. Actually for the neutral excited baryon, this term does not con- For the BBγ and γBB ′ M M ′ couplings we have also the anomalous coupling terms, which are gaugeinvariant by themselves. For these couplings, we use the interaction Lagrangian appearing in the chiral perturbation theory [28]: with the electromagnetic field tensor F µν , the charge matrix Q, the spin matrix S µ and the chiral field (3) matrix of the Nambu-Goldstone boson field. This interaction Lagrangian gives us spatial components of both the BBγ and the γBB ′ M M ′ vertices: − iΓ where we have made non-relativistic reduction. µ (A) i are the anomalous magnetic moments of the baryons, and the matrix A ij is given as, with the coefficients X ij , Y ij fixed only by the flavor SU(3) symmetry and their explicit values are found in Ref. [15]. The anomalous magnetic moments are given by the interaction Lagrangian (16) as µ  (19), while for the baryon magnetic moments we use the experimental values. For the unobserved Σ 0 magnetic moment, we use the SU(3) flavor relation µ Σ 0 = (µ Σ + + µ Σ − )/2, which is consistent with the quark model. We neglect the transition magnetic moment µ ΛΣ 0 , because this interaction changes the isospin of the excited baryons 0 to 1.
After obtaining the elementary couplings, we calculate the amplitudes, T µ γ1ij , T µ γ2ij and T µ γ3ij , accordingly to the Feynman diagrams given in Fig. 2: where the loop integrals are given with the photon couplings to the meson and baryon by The loop function G in Eq. (23) is regularized in the same way as Eq. (3) with the subtraction constants (5). This treatment, in fact, is necessary for the gauge-invariant calculation. The integrals in Eqs. (24) and (25) are convergent and require no regularizations, because V µ M l and V µ B l are not more than second power of the loop momentum.
In the present calculation, we did not introduce the form factors for the ground state mesons and baryons and treat them as point particles, because we are interested in the sizes of the excited baryon generated by meson-baryon dynamics and in estimation of pure dynamical effects. For the qualitative argument or comparison with experiments (if possible), the inclusion of the form factors will be important. In this formulation, inclusion of the form factors for the mesons and baryons is straightforward; we simply multiply the meson and baryon form factors to each vertex. Here, the gauge invariance requires to use a common form factor F (Q 2 ) to every vertex. The common form factor can be factorized out from the loop integrals since it depends only on the photon momentum, so the correction from the inclusion of the form factor is multiplicative: G(Q 2 ) = G(Q 2 )F (Q 2 ), whereG(Q 2 ) and G(Q 2 ) are the form factors of excited baryons with and without the inclusion of the meson and baryon form factors, respectively. It is interesting to note that the electric mean squared radii for neutral excited baryons do not depend on the inclusion of the meson and baryon form factor, since (26) and G E (0) = 0 for neutral excited baryons and F (0) = 1 by definition. Thus, the present results for r 2 E of the Λ(1405) remain unchanged even with inclusion of the meson and baryon form factor.

Numerical results
We show our result for the electromagnetic mean squared radii of the excited baryons in Table 1. As mentioned before, we have two Λ(1405) states, z 1 and z 2 . The lower state strongly couples to the πΣ channel, while the higher state dominantly couples to theKN channel, as one can see from the analysis of g i in Ref. [9]. Since the π + Σ − and π − Σ + contribute to the isospin 0 state almost equally, the electric mean squared radius of the lower Λ(1405) state z 1 is suppressed.
The electric mean squared radius of the higher Λ(1405) state z 2 is more interesting. The higher state has a three times larger absolute value of r 2 E , 0.32 fm 2 , than that of neutron ∼ −0.12 fm 2 . This observation implies that the electric form factor of the Λ(1405) is softer, namely has larger energy de-  pendence, than that of the neutron, since the electric form factors for the neutral particles are given by G E (Q 2 ) = −Q 2 r 2 E /6 + · · · in the expansion of Q 2 due to the neutral charge G E (0) = 0. The softer form factor means that the Λ(1405) is more spatially extended. This is because kaon inside the Λ(1405) has less "virtuality" than pions surrounding the nucleon [29]. The negative charge radius of the neutron is interpreted as distribution of π − cloud. For the neutron case, the pion cloud consists of completely virtual pions, since the system needs at least 140 MeV to create a pion. In contrast, for the z 2 pole of the Λ(1405), only several MeV is required to makē KN state which is the dominant component of this resonance according to the analysis of the coupling strengths [9]. Therefore, the K − inside the Λ(1405) can be largely distributed.
These results are free from the non-resonant background since we calculate the mean squared radii on the top of the Λ(1405) resonance pole in the complex energy plane. Thus, these values have definite theoretical meanings and are used for comparison with other models. On the other hand, experimental observables may be the ratio of the amplitudes T µ γij /T ij in the real energies, which includes the contributions from the non-resonant meson-baryon scattering states [15].
In order to extract the information of the sizes from the form factors, we perform the following analysis. If the decay width of the resonance is small, the imaginary parts of the mean squared radii are small. Thus, it is possible to interpret the mean squared radii as the sizes for the resonant states, since the radii are close to real numbers. For this purpose, we analyze the mean squared radii of a Λ(1405) calculated without the πΣ channel, which is the main decay mode of the actual Λ(1405). Neglecting the couplings to the πΣ channel in C ij and leaving other parameters as in the calculation of the actual Λ(1405), we find the pole position 1422.34 − 0.02i MeV, the mean squared radii r 2 E = −0.519−0.008i fm 2 and We also perform analyses for another interesting system of aKN bound state, which is generated only by the attractive interaction of theKN channel [21]. This bound state is considered to be an origin of the higher state of the actual Λ(1405), z 2 . To investigate this state, we introduce a parameter y (0 ≤ y ≤ 1) in front of the off-diagonal couplings ofKN to the other channels in C ij . The case of y = 1 corresponds to the full coupled-channel calculation, whereas y = 0 corresponds to the calculation with only theKN channels.
The trajectory of the pole position from y = 1 to y = 0 is shown in Fig. 3. We also show the corresponding electromagnetic mean squared radii in Fig. 3. We thus estimate the size of a mesonbaryon resonance by theKN bound state which appears at 1429 MeV in our model (see Tab. 1). The bound state consists of K − p andK 0 n components. The electric mean squared radius reflects the charge distribution of the K − p component, since theK 0 n, in which both hadrons are charge neutral, does not contribute to the electric interactions. The negative sign for the radius implies that the K − is surrounding around the proton. In addition, with the fact that the electromagnetic size of the proton is roughly 0.9 fm, our result of the electric root mean squared radius | r 2 E | ≃ 1.48 fm implies that Λ(1405) has structure of widely spread K − clouds around the core of proton with larger size than that of typical ground state baryons. The magnetic mean squared radius, r 2 M , represents distribution of magnetic moments of the core nucleons in theKN bound system, because anti-kaons cannot contribute to magnetic interactions. Due to dynamics of the nucleon and anti-kaon, the nucleon is distributed inside of the Λ(1405), of which size is seen as the mean squared radii. This implies that the magnetic interaction also suggests a larger size for the Λ(1405) than the ordinary baryons.
The results of the mean squared radii of the Λ(1405) without the πΣ decay channel are much smaller than those of theKN bound state. This is because the former case provides deeper bound state than the latter case. The resonance energies of these cases are very similar as seen by just 7 MeV difference out of about 1 GeV energy scale, but the binding energies have two times difference. To investigate the relation between the radii and the binding energy, we consider theKN bound state and tune the binding energy by changing the subtraction constant aK N , which is the only parameter of this system. In Fig. 4, we show the electric mean squared radius as a function of the mass of theKN bound state. This figure implies that the deeper bound states have the smaller radii. Since the smaller size of the bound state is expected in the deeper bound state, the obtained radius can be interpreted as the size of the Λ(1405). Therefore, knowing precise resonance point of the Λ(1405) is very important to understand its properties.
Let us examine our result of electric mean squared radius in comparison with theoretical estimates of the size of the Λ(1405). In the study of the kaonic nuclei, single-channelKN potentials were constructed from the phenomenological interaction [20] and from the chiral coupled-channel approach [21], both of which reproduce experimental data ofKN phenomenology. The relative mean distance ofK and N in Λ(1405) was estimated to be about 1.36 fm by phenomenological potential [20] and about 1.8 fm by chiral potential [22] where the difference of the results is attributed to the difference of binding energies and strengths of the potentials. Our result, | r 2 E | ≃ 1.48 fm, is quantitatively similar to the estimated size of the Λ(1405), although one should note that we have evaluated the charge radius which cannot be directly compared with the mean distance ofK and N .

Conclusions
We have calculated the electromagnetic mean squared radii of Λ(1405) based on the meson-baryon picture in the chiral unitary model. The evaluation of the electromagnetic mean squared radii have been performed in two ways: In the first approach, with full coupled channels for the Λ(1405), the mean squared radii in complex numbers are obtained at the poles for the physical resonant states and the absolute values suggest that the form factors for the Λ(1405) is softer than those for the neutron. In the second, we describe Λ(1405) as a bound state of KN by neglecting all the off-diagonal couplings of KN to the other channels, in order to estimate the size of the resonant state. As a consequence of the small binding energy in chiral unitary model, our result implies that K − in Λ(1405) is widely spread around p and that the size of the Λ(1405) is larger than that of typical ground state baryons.
In this appendix, we show that the present formulation for the amplitude with electromagnetic coupling developed in Sec. 3 and 4 satisfies Ward-Takahashi identity at the resonance point of the excited baryon with charge Q H .
The photon coupling amplitude in the present formulation is given by three contributions: The definition of each term is, as given in Eqs. (21), (22) and (23), where s ≡ P µ P µ , s ′ ≡ P ′µ P ′ µ and P ′µ = P µ + q µ with the photon momentum q µ . Let us also recall the photon couplings appearing these three contributions: − iΓ µ ij (P, P ′ ) = iQ H C ij 2f 2 P µ + P ′µ s + s ′ , with incoming and outgoing momenta of mesons k µ , k ′µ and baryons p µ , p ′µ , charge of mesons Q Mi and baryons Q B i , and P µ = k µ + p µ , P ′µ = k ′µ + p ′µ , Q H = Q M + Q B . Here we have omitted the terms with the Pauli matrices σ, which vanish when q µ is multiplied.
With the aid of the following identity, we can obtain the loop integrals in Eqs (24) and (25) in terms of the loop function G: Therefore, multiplying q µ to T µ γ1ij + T µ γ2ij , we obtain Here G ′ and T ′ mean G( √ s ′ ) and T ( √ s ′ ), respectively.
The coupling Γ µ ij satisfies with the Weinberg-Tomozawa interaction V ′ ij ≡ V ij ( √ s ′ ) and V ij ≡ V ij ( √ s). Thus, Collecting all the terms and using BS equation (1), we obtain in matrix form. This equation has only single pole terms in the right-hand side. For calculation of the form factor of the baryon resonances, we extract residues of double pole of the amplitude in the Breit frame and the right-hand side of this equation does not contribute to the residues. Therefore, the present formulation satisfies Ward-Takahashi identity at the resonance position and the form factors evaluated by T µ γ are gauge-invariant. It is also seen in the above argument that, off the pole position, it is necessary to include the other terms shown in Ref. [27] for gauge invariance.