Photoproduction of Theta^+(1540,1/2^+) reexamined with new theoretical information
YYITP-09-01INHA-NTG-01/09
Photoproduction of Θ + (1540 , / + ) reexaminedwith new theoretical information Seung-il Nam ∗ and Hyun-Chul Kim † Yukawa Institute for Theoretical Physics (YITP),Kyoto University, Kyoto 606-8502, Japan Department of Physics, Inha University, Incheon 402-751, Korea. (Dated: November 2, 2018)
Abstract
We reinvestigate the photoproduction of the exotic pentaquark baryon Θ + (1540 , / + ) from the γN → ¯ K Θ + reaction process within the effective Lagrangian approach, taking into account newtheoretical information on the KN Θ and K ∗ N Θ coupling strengths from the chiral quark-solitonmodel ( χ QSM). We also consider the crossing-symmetric hadronic form factor, satisfying the on-shell condition as well. Due to the sizable vector and tensor couplings for the vector kaon, g K ∗ N Θ and f K ∗ N Θ , which are almost the same with the vector coupling g KN Θ ≈ . K ∗ -exchange contribution plays a critical role in the photon beam asymmetries. PACS numbers: 13.60.Le, 14.20.JnKeywords: Exotic pentaquark baryon, Θ + photoprodution ∗ E-mail: [email protected] † E-mail: [email protected]
Typeset by REVTEX 1 a r X i v : . [ h e p - ph ] J a n . INTRODUCTION Since the LEPS collaboration announced the evidence of the Θ + [1], being motivated byRef. [2] in which its decay width was predicted to be very small with its mass 1540 MeV [3],it intrigued a great deal of experimental and theoretical works on the Θ + (see, for example,reviews [5, 6] for the experimental and theoretical status before 2006). However, the CLAScollaboration conducted a series of experiments and reported eventually null results of findingthe Θ + [7, 8, 9, 10] in various reactions. Considering the fact that these CLAS experimentswere dedicated ones with high statistics, these null results from the CLAS experiment areremarkable and indicate that the total cross sections for photoproductions of the Θ + shouldbe tiny. In fact, the 95% confidence level (CL) upper limits on the total cross sections for theΘ + at 1540 MeV lie mostly in the range of (0 . − .
8) nb [7, 8, 10]. In Ref. [9] the upper limiton the γd → Λ(1520 , / − )Θ + total cross section turned out to be about 5 nb in the massrange from 1520 MeV to 1560 MeV with a 95 % CL. The KEK-PS-E522 collaboration [11] hascarried out the experiment searching for the Θ + via the π − p → K − X reaction and found abump at around 1530 MeV but with only (2 . ∼ . σ statistical significance. Moreover, theupper limit of the Θ + -production cross section in the π − p → K − Θ + reaction was extractedto be 3 . µ b. A later experiment at KEK (KEK-PS-E559), however, has observed no clearpeak structure for the Θ + in the K + p → π + X reaction [12], giving a 95 % CL upper limitof 3 . µ b/sr for the differential cross section averaged over from 2 ◦ to 22 ◦ in the laboratorysystem. This negative situation is summarized in the 2008 Review of Particle Physics byWohl [4]: “ The whole story – the discoveries themselves, the tidal wave of papers by theoristsand phenomenologists that followed, and the eventual “undiscovery” – is a curious episodein the history of science. ”In the meanwhile, the DIANA collaboration has continued to search for the Θ + in the K + n → K p reaction and has reported a direct formation of a narrow pK peak with massof (1537 ±
2) MeV and width of Γ = (0 . ± .
11) MeV [13]. Compared to the formermeasurement by the DIANA collaboration for the Θ + , the decay width was more preciselymeasured in this new experiment [14], the statistics being doubled. The SVD experimenthas also reported a narrow peak with the mass, (1523 ± stat . ± syst . ) MeV in the inclusivereaction pA → pK s + X [15, 16]. Moreover, the LEPS collaboration has brought news veryrecently on the evidence of the Θ + [17]: The mass of the Θ + is found at (1525 ± . σ . The peak position isshifted by +3 MeV systematically due to the minimum momentum spectator approximation(MMSA). The differential cross section was estimated to be (12 ±
2) nb / sr in the photonenergy ranging from 2.0 GeV to 2.4 GeV in the LEPS angular range.Although it seems that the pentaquark baryon Θ + rarely exists according to certainexperiments as explained above, it is still theoretically necessary to understand why the Θ + is so elusive and intractable. As mentioned previously, one of the reasons can be found in thefact that the cross sections of the Θ + photoproduction as well as of the mesonic productionare observed to be minuscule. The origin of these tiny cross sections can be understoodby the smallness of the KN Θ and K ∗ N Θ coupling constants, as mentioned explicitly inRefs. [12]. A similar conclusion was also found in Ref. [17]. Moreover, the decay width ofthe Θ + → KN , observed by the DIANA collaboration, indicates that the KN Θ couplingconstant must be small, as was reviewed in Ref. [18]. From the theoretical side, Ref. [19] hasshown that the tensor coupling constant for the K ∗ N Θ vertex is very small and has predictedthe total cross section for the Θ + photoproduction to be around 0 . + photoproduction was estimated to bebelow (0 . ∼ .
8) nb [7, 8, 10]. Azimov et al. [20] has evaluated the smaller value ofthe K ∗ N Θ tensor coupling constant, employing the vector meson dominance with SU(3)symmetry. The vector coupling constant for the K ∗ N Θ vertex vanishes in SU(3) symmetrydue to the generalized Ademollo-Gatto theorem as shown in Ref. [21], in which the vectorand tensor coupling constants for the K ∗ N Θ vertex turned out to be very small within theframework of the chiral quark-soliton model ( χ QSM) with SU(3) symmetry breaking effectstaken into account: The vector coupling constant g K ∗ N Θ = 0 . ∼ .
87 and tensor couplingconstant f K ∗ N Θ = 0 . ∼ .
16, respectively. In the same theoretical framework, the KN Θcoupling constant was determined to be g KN Θ = 0 .
83, which leads to the corresponding thedecay width of the Θ + : Γ Θ → NK = 0 .
71 MeV [22]. Note that these results in the χ QSM areall derived without adjusting any parameters.In Refs. [24, 25, 26, 29], the photoproduction of the Θ + was investigated, based oneffective Lagrangian approaches. However, since the coupling constants and cut-off masseswere unknown both experimentally and theoretically, it was very difficult to describe theproduction mechanism of the Θ + without any ambiguity. Thus, in the present work, wewant to reexamine the photoproduction of the Θ + , incorporating the KN Θ and K ∗ N Θcoupling cosntants and cut-off masses from Refs. [21, 22]. The results will be shown thatthe magnitudes of the total cross section and differential cross section are qualitativelycompatible with those of the LEPS and CLAS data.We sketch the structure of the present work as follows: In Section II, we briefly reviewthe coupling constants of the K and K ∗ exchange, which play critical roles in describing thephotoproduction of the Θ + . In Section III, we explain the general formalism of the effectiveLagrangian method. In Section IV, we present the numerical results and discuss them. Thefinal Section is devoted to summary and conclusion. II. COUPLING CONSTANTS AND FORM FACTORS FOR THE K AND K ∗ EX-CHANGES FROM THE χ QSM
In this Section, we briefly review the results of the χ QSM calculations. We start with thefollowing Θ + -to-neutron transition matrix elements of the vector current V µ = ¯ ψγ µ ( λ − iλ ) ψ , and axial-vector current A µ = ¯ ψγ µ γ ( λ − iλ ) ψ : (cid:104) Θ( p (cid:48) ) | V µ (0) | n ( p ) (cid:105) = ¯ u Θ ( p (cid:48) ) (cid:20) F n Θ1 ( Q ) γ µ + F n Θ2 ( Q ) iσ µν q ν M Θ + M n + F n Θ3 ( Q ) q µ M Θ + M n (cid:21) u n ( p ) , (1) (cid:104) Θ( p (cid:48) ) | A µ (0) | n ( p ) (cid:105) = ¯ u Θ ( p (cid:48) ) (cid:2) G n Θ1 ( Q ) γ µ + G n Θ2 ( Q ) q µ + G n Θ3 ( Q ) P µ (cid:3) γ u n ( p ) , (2)where u Θ( n ) denotes the spinor of the Θ + (neutron) with the corresponding mass M Θ( n ) . The Q stands for the momentum transfer Q = − q = − ( p (cid:48) − p ) and P represents the totalmomentum P = p (cid:48) + p . F n Θ i and G n Θ i designate real transition form factors, related to thestrong coupling constants for the K ∗ and K with the help of the vector-meson dominance(VMD) [31, 32] and Goldberger-Treiman relations.In the VMD, the vector-transition current can be expressed as the K ∗ current by thecurrent field identity (CFI): V µ ( x ) = ¯ s ( x ) γ µ u ( x ) = m K ∗ f K ∗ K ∗ µ ( x ) , (3)3here m K ∗ and f K ∗ denote, respectively, the mass of the K ∗ meson, m K ∗ = 892 MeV, anddecay constant defined as f K ∗ = m K ∗ m ρ f ρ , (4)where the decay constant f ρ for the rho meson can be determined as f ρ = 4 πα m ρ ρ → e + e − . (5)Here, α denotes the electromagnetic fine-structure constant. The f K ∗ is determined by usingthe ρ -meson experimental data with m ρ = 770 MeV and Γ ρ → e + e − = (7 . ± .
11) keV [4],for which we get the values f ρ ≈ .
96 and f K ∗ ≈ .
71. Then, using the CFI, we can expressthe K ∗ N Θ vertex in terms of the transition form factors in Eqs. (1) and (2): (cid:104) Θ( p (cid:48) ) | ¯ sγ µ u | n ( p ) (cid:105) = m K ∗ f K ∗ m K ∗ − q (cid:104) Θ( p (cid:48) ) | K ∗ µ | n ( p ) (cid:105) , (6) (cid:104) Θ( p (cid:48) ) | K ∗ µ | n ( p ) (cid:105) = ¯ u Θ ( p (cid:48) ) (cid:20) g K ∗ n Θ γ µ + f K ∗ n Θ iσ µν q ν M Θ + M n + s K ∗ n Θ q µ M Θ + M n (cid:21) u n ( p ) , (7)where the g K ∗ n Θ and f K ∗ n Θ denote the vector and tensor coupling constants for the K ∗ N Θvertex, respectively. By comparing the Lorentz-structures the strong coupling constants canbe determined as g K ∗ n Θ = f K ∗ F Θ n (0) , f K ∗ n Θ = f K ∗ F Θ n (0) . (8)Using the generalized Goldberger-Treiman relation, we can get the strong coupling con-stant g Kn Θ for the KN Θ vertex as follows: g Kn Θ = G Θ n (0) ( M Θ + M n )2 f K , (9)where f K ≈ . f π stands for the kaon decay constant.The form factors F n Θ1 ( Q ), F n Θ2 ( Q ) and G Θ nA ( Q ) of Eqs. (1) and (2) can be expressedin terms of the matrix elements of the vector and axial-vector currents with their time andspace components decomposed in the Θ + rest frame as follows: G n Θ E ( Q ) = (cid:90) d Ω q π (cid:104) Θ( p (cid:48) ) | V (0) | n ( p ) (cid:105) , (10) G n Θ M ( Q ) = 3 M n (cid:90) d Ω q π q i (cid:15) ik i q (cid:104) Θ( p (cid:48) ) | V k (0) | n ( p ) (cid:105) , (11) G n Θ A ( Q ) = − q (cid:114) M Θ E Θ + M Θ (cid:90) d Ω q π (cid:104) q × (cid:16) q × (cid:104) Θ( p (cid:48) ) | A (0) | n ( p ) (cid:105) (cid:17)(cid:105) z , (12)where the electromagnetic-like form factors G n Θ E and G n Θ M are written as G n Θ E ( Q ) = (cid:114) E n + M n M n (cid:20) F n Θ1 ( Q ) − F n Θ2 ( Q ) M Θ + M n q E n + M n + F n Θ3 ( Q ) q M Θ + M n (cid:21) , (13) G n Θ M ( Q ) = (cid:114) M n E n + M n (cid:2) F n Θ1 ( Q ) + F n Θ2 ( Q ) (cid:3) . (14)4ince the second and third parts in Eq. (13) turn out to be very small, we take the followingexpressions as the vector and tensor coupling constants: g K ∗ n Θ = f K ∗ G Θ nE (0) , f K ∗ n Θ = f K ∗ ( G Θ nE (0) − G Θ nM (0)) . (15)The next step is to evaluate the form factors of Eqs. (10), (11), and (12) within theself-consistent χ QSM. The model is featured by the following effective low-energy partitionfunction with quark fields ψ with the number of colors N c and the pseudo-Goldstone bosonfield U ( x ) in Euclidean space: Z χ QSM = (cid:90) D ψ D ψ † D U exp (cid:20) − (cid:90) d xψ † iD ( U ) ψ (cid:21) = (cid:90) D U exp( −S eff [ U ]) , (16) S eff ( U ) = − N c Tr ln iD ( U ) , (17)where D ( U ) = γ ( i / ∂ − ˆ m − M U γ ) = − i∂ + h ( U ) − δm, (18) δm = m s − ¯ m γ × + ¯ m − m s √ γ λ = M γ × + M γ λ . (19)The current quark mass matrix is defined as ˆ m = diag( ¯ m, ¯ m, m s ) = ¯ m + δm . The ¯ m desig-nates the average of the up and down current quark masses with isospin symmetry assumed.The M denotes the constituent quark mass of which the best value for the numerical resultsis M = 420 MeV. The pseudo-Goldstone boson field U γ is defined as U γ = exp( iγ λ a π a ) = 1 + γ U + 1 − γ U † (20)with U = exp( iλ a π a ). For the quantization, we consider here Witten’s embedding of SU(2)soliton into SU(3): U SU(3) = (cid:18) U SU(2)
00 1 (cid:19) (21)with the SU(2) hedgehog chiral field U SU(2) = exp[ iγ ˆ n · τ P ( r )] , (22)Here, the P ( r ) denotes the profile function of the chiral soliton U SU(2) .In order to describe the baryonic properties, we first have to derive the profile function.It can be obtained by the following procedure: First, we take the large N c limit and solveit in the saddle-point approximation, which corresponds at the classical level to finding theprofile function P ( r ) in Eq. (22). Thus, the P ( r ) can be obtained by solving numerically thecalssical equation of motion coming from δS eff /δP ( r ) = 0, which yields a classical soliton field U c constructed from a set of single quark energies E n and corresponding states | n (cid:105) related tothe eigenvalue equation h ( U ) | n (cid:105) = E n | n (cid:105) . However, the classical soliton does not have thequantum number of the baryon states, so that we need to project it to physical baryon statesby the semiclassical quantization of the rotational and translational zero modes. Note that5he zero modes can be treated exactly within the functional integral formalism by introducingcollective coordinates. Detailed formalisms can be found in Refs. [33, 34]. Considering therigid rotations and translations of the classical soliton U c , we can express the soliton field as U ( x , t ) = A ( t ) U c ( x − z ( t )) A † ( t ) , (23)where A ( t ) denotes a unitary time-dependent SU(3) collective orientation matrix and z ( t )stands for the time-dependent displacement of the center of mass of the soliton in coordinatespace.In the χ QSM, the baryon state consists of N c valence quarks expressed as | B ( p ) (cid:105) = lim x →−∞ √Z e ip x (cid:90) d (cid:126)x e i (cid:126)p · (cid:126)x J † B ( x ) | (cid:105) (24)with the barynic current: J B ( x ) = 1 N c ! Γ α ··· α Nc B ε i ··· i Nc ψ α i ( x ) · · · ψ α Nc i Nc ( x ) , (25)where α , · · · , α N c and i , · · · , i N c denote the spin-flavor and color indices, respectively. TheΓ α ··· α Nc B stands for the projection operator for the corresponding baryon state. Thus, thetransition matrix elements in Eqs. (10), (11), and (12) can be written as the followingcorrelation functions: (cid:104) B ( p ) |J µχ (0) | B ( p (cid:105) = 1 Z lim T →∞ e − ip T + ip T (cid:90) d (cid:126)x (cid:48) d (cid:126)xe i(cid:126)p · (cid:126)x − i(cid:126)p · (cid:126)x (cid:48) × (cid:90) D U D ψ † D ψJ B (cid:48) (cid:18) T , (cid:126)x (cid:48) (cid:19) J µχ (0) J † B (cid:18) − T , (cid:126)x (cid:19) exp (cid:20) − (cid:90) d x ψ † iD ( U ) ψ (cid:21) . (26)We can solve Eq. (26) in the saddle-point approximation justified in the large N c limit, takinginto account the zero-mode quantization explained before. We consider only the rotational1 /N c corrections and linear m s corrections. Thus, we expand the quark propagators inEq. (26) with respect to Ω and δm to the linear order and ˙ T † z ( t ) T z ( t ) to the zeroth order.Having carried out a tedious but straightforward calculation (see Refs. [33, 34] for details),we finally can express the baryonic matrix elements in Eqs. (10), (11), and (12) as a Fouriertransform in terms of the corresponding quark densities and collective wave-functions of thebaryons: (cid:104) B (cid:48) ( p (cid:48) ) |J χµ (0) | B ( p ) (cid:105) = (cid:90) dA (cid:90) d z e i q · z Ψ ∗ B (cid:48) ( A ) F χµ ( z )Ψ B ( A ) , (27)where Ψ( A ) denote the collective wavefunctions and F χµ represents the quark densities cor-responding to the current operator J χµ . Using the collective wavefunctions and the quarkdensities, we immediately obtain the transition vector and axial-vector form factors G n Θ E,M,A in Eqs. (10), (11), and (12). The corresponding results can be found in Refs. [21, 22], whichare summarized in Table I. The vector coupling constant g K ∗ n Θ vanishes in SU(3) symmetriccase because of the generalized Ademollo-Gatto theorem. Note that even the SU(3) sym-metry breaking effects from the Hamiltonian does not contribute to the g K ∗ n Θ . The value of g K ∗ n Θ with SU(3) symmetry breaking comes solely from the wavefunction corrections. Thecoupling constants for the proton can be obtained easily by considering isospin factors.6ote that there is a sign difference in the coupling constants for the neutron and proton: g K ∗ n Θ = − g K ∗ p Θ and the same for the f K ∗ N Θ [22]. However, as shown in the next section,since the K ∗ -exchange contribution in the Θ + -photoproduction provides a 90 ◦ phase differ-ence from others, these sign differences for the neutron and proton targets do not make anydifference at all in describing physical observables. III. AN EFFECTIVE LAGRANGIAN APPROACH
We now proceed to calculate the amplitudes for the reaction of the Θ + photoproduction,taking the results of the coupling constants in Section II as numerical inputs. We firstdefine the relevant effective interactions to compute the Θ + photoproduction in the Bornapproximation. Since the coupling constants derived from the χ QSM are for the Θ + withpositive parity, we assume here the parity of the Θ + to be positive. L KN Θ = − ig KN Θ ¯Θ γ KN + h . c ., L K ∗ N Θ = − g K ∗ N Θ ¯Θ γ µ K ∗ µ N − f K ∗ N Θ M Θ + M N ¯Θ σ µν ∂ ν K ∗ µ N + h . c ., L γKK = ie K [( ∂ µ K † ) K − ( ∂ µ K ) K † ] A µ + h . c ., L γKK ∗ = g γKK ∗ (cid:15) µνσρ ( ∂ µ A ν )( ∂ σ K † ) K ∗ ρ + h . c ., L γNN = − e N ¯ N (cid:20) γ µ − κ N M N σ µν F µν (cid:21) N + h . c ., L γ ΘΘ = − e Θ ¯Θ (cid:20) γ µ − κ Θ M Θ σ µν F µν (cid:21) Θ + h . c ., (28)where Θ, N , K , and K ∗ µ denote the fields of the Θ + , the nucleon, the pseudoscalar kaon,and the vector kaon, respectively. The A µ represents the photon field, whereas the F µν theantisymmetric electromagnetic field strength. The g γKK ∗ designates the γKK ∗ couplingconstant that can be determined by using the experimental data for the K ∗ → Kγ decay,which yields 0 .
388 GeV − for the neutral decay and 0 .
254 GeV − for the charged one. The e K , e N , and e Θ are unit electric charges for the kaon, nucleon, and Θ + , respectively. The κ N and κ Θ stand for the anomalous magnetic moments of the nucleon and Θ + , respectively.Since the magnetic moment of the Θ + is theoretically known to be rather small [6, 35, 36, 37],we take it to be κ Θ ≈ − . σ µν is a usual antisymmetric spin operator σ µν = i [ γ µ , γ ν ] /
2. The M N and M Θ correspond to the nucleon and Θ + masses, and aretaken to be 939 MeV and 1540 MeV, respectively.Having performed straightforward manipulations, we arrive at the following invariantamplitudes for the s - and u -channel contributions, and the K - and K ∗ -exchange ones in the m s = 0 m s = 180 MeV g K ∗ N Θ f K ∗ N Θ g KN Θ g K ∗ N Θ f K ∗ N Θ g KN Θ .
91 1 .
41 0 .
81 0 .
84 0 . K ∗ N Θ and KN Θ + coupling constants at Q = 0 with and without m s corrections. The constituent quark mass M is taken to be M = 420 MeV. -channel as follows: i M s = − g KN Θ ¯ u ( p ) (cid:20) e N γ F c (/ p + M N ) + F s / k s − M N / (cid:15) − e Q κ N M N γ F s (/ k + / p + M N ) s − M N / (cid:15) / k (cid:21) u ( p ) ,i M u = − g KN Θ ¯ u ( p ) (cid:20) e Θ / (cid:15) F c (/ p + M Θ ) − F u / k u − M γ − e Q κ Θ M Θ / (cid:15) / k F u (/ p − / k + M Θ ) u − M γ (cid:21) u ( p ) ,i M Kt = 2 e K g KN Θ ¯ u ( p ) γ ( k · (cid:15) ) t − M K u ( p ) F c , M K ∗ t = ig γKK ∗ (cid:15) µνσρ ¯ u ( p ) (cid:34) g K ∗ N Θ ( t ) k µ (cid:15) ν k σ γ ρ t − M K ∗ − f K ∗ N Θ ( t )2( M N + M Θ ) k µ (cid:15) ν k σ γ ρ / q t − / q t k µ (cid:15) ν k σ γ ρ t − M K ∗ (cid:35) u ( p ) F / v , (29)where ¯ u and u are the Dirac spinors of Θ + and the nucleon. The four momenta p , p , k , and k stand for those of the nucleon, the Θ + , the photon, and the kaon, respectively.The s , u and t represent the Mandelstam variables. Note that in the case of the process γp → ¯ K Θ + , there is no contribution from the K -exchange contribution. Moreover, the K ∗ -exchange contribution gives a 90 ◦ phase difference from others as mentioned previously.The form factors as functions of the Mandelstem variables are defined as follows [27, 28]: F s,u,t = Λ Λ + [( s, u, t ) − M s,u,t ] , F v = Λ Λ + ( t − M K ∗ ) , (30)where the four-dimensional cutoff mass Λ is chosen to be 650 MeV which is compatible withthose used in the Λ(1520) [29] and Λ(1116) [30] photoproductions. The common overallform factor F c is written as follows: F c ( s, t, u ) = 1 − (1 − F s )(1 − F u )(1 − F t ) , (31)which satisfies the on-shell condition and crossing symmetry. We note that this form-factorscheme preserves the Ward-Takahashi identity explicitly.As for the coupling constants g K ∗ N Θ , f K ∗ N Θ , and g KN Θ , we take the forms of Eq. (15)with the results of the χ QSM. The corresponding form factors from the χ QSM can beparameterized as follows [21]: G n Θ E ( t ) = G E (cid:20) α E Λ E α E Λ E − t (cid:21) α + b, G n Θ M ( t ) = G M (cid:20) α M Λ M α M Λ M − t (cid:21) α , (32)where fitting parameters α E,M , Λ
E,M , and b are listed in Table II. These parameters aredetermined by using the results with the strange quark mass m s = 180 MeV and theconstituent quark mass M = 420 MeV. IV. NUMERICAL RESULTS AND DISCUSSION
We are now in a position to discuss the results of the present work. As mentionedpreviously, we assume that the Θ + baryon has the spin-parity quantum number, 1 / + . In8he left panel of Fig. 1, each contribution to the total cross section of the γn → K − Θ + reaction is drawn as a function of the photon energy E γ . As shown there in Fig. 1, the K -exchange contribution is the most dominant one, whereas the second dominant one comesfrom the K ∗ -exchange one. The K ∗ -exchange contribution is, however, the most dominantone near the threshold up to around 1 . u -channel contribution is rather small and the s -channel is almost negligible due to the present form-factor scheme that suppresses thesechannels. The K -exchange contribution shows rather strong dependence on the photonenergy E γ . As the E γ increases, it starts to get enhanced.The right panel of Fig. 1 depicts each contribution to the differential cross section of the γn → K − Θ + reaction. The tendency of each contribution is in general similar to the case ofthe total cross section, i.e. the K -exchange contribution turns out to be the dominant one,as it should be. Moreover, it has a large bump structure in the forward direction. Beginningfrom the backward direction, it is getting increased slowly. In the forward direction, it startsto increase drastically till around cos θ cm ≈ .
75 and falls down sharply, which makes the K -exchange contribution have the bump structure. The K ∗ -exchange contribution is alsolarger in the forward direction than in the backward direction. The u -channel contributesmainly to the backward direction as expected. Summing up all contributions, we can easilysee that the differential cross section is much more enhanced in the forward direction. A. Effects of the K ∗ exchange Since the present work is mainly interested in the effects of the K ∗ -exchange contribution,we now examine the features of the K ∗ -exchange contribution to various observable in detail.In the two-upper panels of Fig. 2, the total cross sections of the γn → K − Θ + and γp → ¯ K Θ + reactions are drawn in the left and right panels, respectively. The K ∗ -exchangecontribution turns out to be almost 30 % to the total cross section for the neutron target.On the other hand, it is almost everything for the proton target. It can be easily understoodfrom the fact that for the γp → ¯ K Θ + reaction there is no K -exchange contribution that isdominant in the neutron channel. Comparing the total cross sections for the neutron targetwith the proton one, we find that that for the neutron one is about 30 % larger than thatfor the proton one, although they are qualitatively in a similar order < ∼ γn → K − Θ + and γp → ¯ K Θ + reactions with and without the K ∗ -exchange contribution for threedifferent photon energies 2 . . . K - and K ∗ -exchange contributions, one can observe the bumpstructures in the region < ∼ ◦ for both the neutron and proton target cases. As in the caseof the total cross sections, while the K ∗ -exchange contribution makes the differential crosssection about 10 % enhanced for the neutron target, its effects are remarkably large for the G n Θ E ( t ) G n Θ M ( t ) G E α E Λ E b G M α M Λ M .
182 9 .
01 0 . − .
04 0 .
286 0 .
851 0 . t . The general tendency is very similar to that of the differentialcross sections shown in Fig. 2. It is worth mentioning that the best way to examine theeffects of the K ∗ -exchange contribution is to investigate the photon beam asymmetry, sincethe K ∗ meson is a vector meson which manifests magnetic meson-baryon coupling behaviorin the present photoprodcution process. The photon beam asymmetry is defined asΣ = (cid:20) dσd Ω ⊥ − dσd Ω (cid:107) (cid:21) × (cid:20) dσd Ω ⊥ + dσd Ω (cid:107) (cid:21) − , (33)where the subscript ⊥ ( (cid:107) ) denotes that the polarization vector of the incident photon isperpendicular (parallel) to the reaction plane. In the two-lower panels of Fig. 3 draws thephoton beam asymmetries for the γn → K − Θ + and γp → ¯ K Θ + reactions in the left andright panels, respectively. When we switch off the K ∗ -exchange contribution, the photonbeam asymmetry for the neutron target, starting from the backward direction, is broughtdown drastically and reaches down to almost Σ = − θ cm = 90 ◦ , due to theelectric meson-baryon coupling of the dominant K -exchange contribution. However, whenwe turn on the K ∗ -exchange one, the photon beam asymmetry decreases mildly from thebackward direction to the forward direction, and then it increases sharply to Σ = 0. On thewhole, the photon beam asymmetry is negative for the neutron target.When it comes to the proton target, the K ∗ -exchange contribution shows profound effectson the photon beam asymmetry. While the photon beam asymmetry becomes negativewithout the K ∗ -exchange contribution, it turns into being positive with it in all the regions.With the K ∗ -exchange contribution switched on, the photon beam asymmetry starts toincrease from the bakcward direction to the forward direction, and it goes down from aroundcos θ cm = 0 . B. Effects of explicit SU(3) symmetry breaking
In Sec. II, it was mentioned that the vector coupling constant g K ∗ N Θ vanishes in theSU(3) symmetric case due to the generalized Ademollo-Gatto theorem. Moreover, the tensorcoupling constant f K ∗ N Θ is also very sensitive to SU(3) symmetry breaking as shown inTable I. Thus, it is of great interest to see the effects of SU(3) symmetry breaking on the Θ + photoproduction. In Fig. 4, we show the total (upper), differential (middle) cross sections,and photon beam asymmetries (lower) for the neutron and proton targets in the left andright panels, respectively. Although the values of the vector and tensor couplings for the K ∗ N Θ vertex are rather sensitive to the effects of SU(3) symmetry breking, all the resultsindicate that SU(3) symmetry breaking does not play any significant role in describing theΘ + photoproduction. The reason can be found in the fact that while the tensor couplingconstant f K ∗ N Θ is almost three times reduced by SU(3) symmetry breaking, the vectorcoupling constant g K ∗ N Θ comes solely from the wavefunction corrections that are also a partof the SU(3) symmetry breaking effects. Thus, the finite value of the g K ∗ N Θ makes up forthe reduction of the f K ∗ N Θ , so that the effects of SU(3) stmmetry breaking turn out to berather small. 10 . SUMMARY AND CONCLUSION In the present work, we have investigated the Θ + photoproduction, taking the new resultsof the chiral quark-soliton model [21, 22] into account. We first have briefly reviewed theformalism of the chiral quark-soliton model for deriving the coupling constants and formfactors for the K ∗ N Θ and KN Θ vertices. We made use of these coupling constants andform factors as numerical inputs for calculating the γn → K − Θ + and γp → ¯ K Θ + scatteringprocesses.We have examined the effects of each contribution to the total and differential crosssections. It turned out that the K -exchange contribution is the most dominant one exceptfor the near-threshold region in which the K ∗ -exchange contributes mainly. The differentialcross section has a large bump structure in the forward direction, which is obviously due tothe K - and K ∗ -exchange contributions.Since it is of great importance to understand how the K ∗ -exchange contribution plays arole in the Θ + photoproduction, we thoroughly have studied the effects of the K ∗ -exchangecontribution in various observables for the γn → K − Θ + and γp → ¯ K Θ + reactions. Itturned out that the K ∗ -exchange contribution is in general the most dominant one in the γp → ¯ K Θ + reaction, since there is no K -exchange contribution for the proton target. The K ∗ -exchange contribution to the γn → K − Θ + reaction is in general about 30 % but to the γp → ¯ K Θ + reaction it is almost everything, since the K -exchange contribution is absent inthis case. In order to see the effects of the K ∗ -exchange contribution, we also calculated thephoton beam asymmetries. It was shown that with the K ∗ -exchange contribution the photonbeam asymmetries are very different from those without the K ∗ -exchange contribution. Inparticular, the photon beam asymmetry for the proton target is changed from the negativesign to the positive with K ∗ -exchange turned on.The effects of SU(3) symmetry breaking are remarkable on the coupling constants forthe K ∗ N Θ vertex. The vector coupling constant g K ∗ N Θ vanishes without SU(3) symmetrybreaking, i.e. g K ∗ N Θ = 0. The tensor coupling constant becomes f K ∗ N Θ = 2 .
91 that is byno means small. When we switched on SU(3) symmetry breaking, the vector-coupling con-stant became finite because of the wavefunction corrections. Moreover, the tensor couplingconstant is reduced about by a factor of three. Thus, one may see this large changes inthe coupling constants in the observables. However, it was found that the effects of SU(3)symmetry breaking were rather small in all observables we calculated in the present work.It indicates that the finite value of the vector coupling constant makes up for the reductionof the tensor coupling constant. Thus, the effects of SU(3) symmetry breaking altogetherturn out to be small.Recent KEK and LEPS experiments [12, 17] have drawn a conclusion that the K ∗ N Θcoupling constants must be small, since the total cross sections turned out to be tiny for theproton target, being rather different from the present results. In contrast, the present resultsfor the neutron target is compatible with the experimental data [17]. In order to pin downthis inconsistency between the theory and experiments, it is of great importance to have morehigh-statistics data for various physical observable for the Θ + -production experiment [38]. Acknowledgment
The authors would like to thank J. K. Ahn, A. Hosaka, and T. Nakano for fruitful dis-cussions. They especially are grateful to T. Ledwig for discussing the results from the chiral11uark-soliton model. The work of S.i.N. is partially supported by the Grant for ScientificResearch (Priority Area No.17070002 and No.20028005) from the Ministry of Education,Culture, Science and Technology (MEXT) of Japan. The work of H.Ch.K. is supportedby the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2006-312-C00507). This work was done under the Yukawa International Program forQuark-Hadron Sciences. The numerical calculations were carried out on MIHO at RCNP inOsaka University and YISUN at YITP in Kyoto University. [1] T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. , 012002 (2003).[2] D. Diakonov, V. Petrov, and M. V. Polyakov, Z. Phys. A , 305 (1997).[3] M. Prasza(cid:32)lowicz, Phys. Lett. B , 234 (2003).[4] C. Amsler et al. , Phys. Lett. B667, 1 (2008)[5] K. H. Hicks, Prog. Part. Nucl. Phys. , 647 (2005) and references therein.[6] K. Goeke et al. , Prog. Part. Nucl. Phys. , 350 (2005).[7] M. Battaglieri et al. [CLAS Collaboration], Phys. Rev. Lett. , 042001 (2006).[8] B. McKinnon et al. [CLAS Collaboration], Phys. Rev. Lett. , 212001 (2006).[9] S. Niccolai et al. [CLAS Collaboration], Phys. Rev. Lett. , 032001 (2006).[10] R. De Vita et al. [CLAS Collaboration], Phys. Rev. D , 032001 (2006).[11] K. Miwa et al. [KEK-PS E522 Collaboration], Phys. Lett. B , 72 (2006).[12] K. Miwa et al. , Phys. Rev. C , 045203 (2008).[13] V. V. Barmin et al. [DIANA Collaboration], Phys. Atom. Nucl. , 35 (2007).[14] V. V. Barmin et al. [DIANA Collaboration], Phys. Atom. Nucl. , 1715 (2003).[15] A. Aleev et al. [SVD Collaboration], hep-ex/0509033.[16] A. Aleev et al. [SVD Collaboration], arXiv:0803.3313 [hep-ex].[17] T. Nakano et al. [LEPS collaboration], arXiv:0812.1035 [nucl-ex].[18] D. Diakonov, Prog. Theor. Phys. Suppl. , 1 (2007).[19] H. Kwee, M. Guidal, M. V. Polyakov and M. Vanderhaeghen, Phys. Rev. D , 054012 (2005).[20] Y. Azimov, V. Kuznetsov, M. V. Polyakov and I. Strakovsky, Phys. Rev. D , 054014 (2007).[21] T. Ledwig, H. -Ch. Kim and K. Goeke, Nucl. Phys. A , 353 (2008).[22] T. Ledwig, H. -Ch. Kim and K. Goeke, Phys. Rev. D , 054005 (2008).[23] S. i. Nam, A. Hosaka and H. -Ch. Kim, Phys. Lett. B , 43 (2004).[24] W. Liu and C. M. Ko, Nucl. Phys. A , 215 (2004).[25] Y. s. Oh, H. c. Kim and S. H. Lee, Phys. Rev. D , 014009 (2004).[26] S. i. Nam, A. Hosaka and H. -Ch. Kim, J. Korean Phys. Soc. , 1928 (2006).[27] H. Haberzettl, C. Bennhold, T. Mart and T. Feuster, Phys. Rev. C , 40 (1998).[28] R. M. Davidson and R. Workman, Phys. Rev. C , 025210 (2001).[29] S. i. Nam, A. Hosaka and H. -Ch. Kim, Phys. Rev. D , 114012 (2005).[30] S. Ozaki, H. Nagahiro and A. Hosaka, Phys. Lett. B , 178 (2008).[31] J. J. Sakurai, Annals Phys. , 1 (1960); “Currents and Mesons”,(The University of Chicago Press, Chicago, 1969).[32] R. P. Feynman, “Photon-Hadron Interactions,” (W.A. Benjamin, Inc. Reading, MA, 1972).[33] C. V. Christov et al., Prog. Part. Nucl. Phys. , 91 (1996).[34] H. -Ch. Kim, A. Blotz, M.V. Polyakov and K. Goeke, Phys. Rev. D , 4013 (1996).[35] H. -Ch. Kim and M. Praszalowicz, Phys. Lett. B , 99 (2004).
36] Gh. S. Yang, H. -Ch. Kim, M. Praszalowicz and K. Goeke, Phys. Rev. D , 114002 (2004).[37] P. Z. Huang, W. Z. Deng, X. L. Chen and S. L. Zhu, Phys. Rev. D , 074004 (2004).[38] Private communications with T. Nakano and H. Kohri for the LEPS collaboration at SPring-8. n → K − Θ + (1 / + )0.000.250.500.75 1.8 2.0 2.2 2.4 2.6 E γ [GeV] σ [ nb ] K ∗ -exchange ............. ............. ............. ........................ ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....... s -channel ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. K -exchange ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ u -channel .......................... ... .................................................... ... .......................... ... .......................... ... .......................... ... .......................... ... .......................... ... .......................... ... .......................... ... .......................... ... .......................... ... .......................... ... ..... γn → K − Θ + (1 / + ) at E cm = 2 . − . − . θ cm d σ / d c o s θ c m [ nb ] K -exchange ........................................................................ ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... K ∗ -exchange ............. ............. ............. .......................................................................................................................................................................................................................................................................................................................................................................................... s -channel ... ... ... ... ... ... ........................................................................................................................................................................... u -channel .......................... ... ................... ................................................................................................................................................................................................................................................................................................................................................................................................................... FIG. 1: Each contribution to the total and differential cross sections for the γn → K − Θ + reaction.The total cross section is drawn in the left panel, while the differential cross section for the photonenergy E γ = 2 . K -exchange contribution,the dashed one for the K ∗ -exchange, the dotted curve for the s -channel, and the dash-dotted onefor the u -channel contributions. n → K − Θ + (1 / + )0.00.51.01.5 1.8 2.0 2.2 2.4 2.6 E γ [GeV] σ [ nb ] With K ∗ .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. Without K ∗ ............. ............. ............. ........................ ............. ............................................................................................................................................................................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. γp → ¯ K Θ + (1 / + )0.00.51.01.5 1.8 2.0 2.2 2.4 2.6 E γ [GeV] σ [ nb ] With K ∗ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... Without K ∗ ............. ............. ............. ........................ ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .... γn → K − Θ + (1 / + )0.00.51.01.5 − . − . θ cm d σ / d c o s θ c m [ nb ] K ∗ ........................................................................ ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... without K ∗ ............. ............. ............. 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γp → ¯ K Θ + (1 / + )0.00.51.01.5 − . − . θ cm d σ / d c o s θ c m [ nb ] K ∗ ........................................................................ ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... without K ∗ ............. ............. ............. 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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. FIG. 2: Effects of the K ∗ -exchange on the total (upper panels) and differential (lower panels) crosssections. The left panels represent those for the γn → K − Θ + reaction, while the right panels thosefor the γp → ¯ K Θ + . The solid curves indicate those with all contributions, whereas the dashedone those without the K ∗ -exchange. The differential cross sections are drawn for three differentphoton energies E γ , 2 . . . n → K − Θ + (1 / + )0.51.01.50.0 0.5 1.0 1.5 2.0 − t [GeV ] d σ / d t [ nb / G e V ] K ∗ ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ without K ∗ ............. ............. ............. ................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. 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γp → ¯ K Θ + (1 / + )0.51.01.50.0 0.5 1.0 1.5 2.0 − t [GeV ] d σ / d t [ nb / G e V ] K ∗ ................................................................................................................................................................................................................................................................................................................................................................................................... without K ∗ ............. ............. ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ γn → K − Θ + (1 / + ) − . − . − . − . θ cm Σ with K ∗ ........................................................................ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ without K ∗ ............. ............. ............. 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γp → ¯ K Θ + (1 / + ) − . − . − . − . θ cm Σ K ∗ ........................................................................ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... without K ∗ ............. ............. ............. 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FIG. 3: Effects of the K ∗ -exchange on the t -dependences ( dσ/dt , upper panels) and the photon-beam asymmetries (Σ, lower panels). The left panels represent those for the γn → K − Θ + reaction,while the right panels those for the γp → ¯ K Θ + . The solid curve indicates the case with allcontributions, whereas the dashed one that without the K ∗ -exchange. The curves are drawn forthree different photon energies E γ , 2 . . . n → K − Θ + (1 / + )0.00.51.01.5 1.8 2.0 2.2 2.4 2.6 E γ [GeV] σ [ nb ] SU(3) symmetry breaking ..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
SU(3) ............. ............. ............. ........................ ............. ................................................................................................................................................................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ......... γp → ¯ K Θ + (1 / + )0.00.51.01.5 1.8 2.0 2.2 2.4 2.6 E γ [GeV] σ [ nb ] SU(3) symmetry breaking ...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
SU(3) ............. ............. ............. ........................ ............. ............................................................................................................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. γn → K − Θ + (1 / + )0.00.51.01.5 − . − . θ cm d σ / d c o s θ c m [ nb ] SU(3) symmetry breaking ........................................................................ ...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
SU(3) ............. ............. ............. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ γp → ¯ K Θ + (1 / + )0.00.51.01.5 − . − . θ cm d σ / d c o s θ c m [ nb ] SU(3) symmetry breaking ........................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
SU(3) ............. ............. ............. ................................................................................................................................................................................................................................................................................................................................................................................................................... γn → K − Θ + (1 / + ) − . − . − . − . θ cm Σ SU(3) symmetry breaking ........................................................................ ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
SU(3) ............. ............. ............. .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... γp → ¯ K Θ + (1 / + ) − . − . − . − . θ cm Σ SU(3) symmetry breaking ........................................................................ ..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
SU(3) ............. ............. ............. ......................................................................................................................................................................................................................................................................................................................................................................................................
FIG. 4: Effects of SU(3) symmetry breaking on the total ( σ , upper), differential ( dσ/d cos θ cm ,middle) cross sections, and photon-beam asymmetries (Σ, lower) as functions of the photon energy E γ . The left panels represent those for the γn → K − Θ + reaction, while the right panels those forthe γp → ¯ K Θ + . The solid curve draws the total cross section with SU(3) symmetry breaking,whereas the dashed one depicts that without it. The curves are drawn for three different photonenergies E γ , 2 . . .3 GeV.