parity determination in the reaction

PHYSICAL REVIEW C 71, 035203 (2005)

Some aspects of + parity determination in the reaction γ N → + K → NKK

A. I. Titov,1,2 H. Ejiri,3,4 H. Haberzettl,5,6 and K. Nakayama5,7

1Advanced Photon Research Center, Japan Atomic Energy Research Institute, Kizu, Kyoto, 619-0215, Japan2Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia

3Natural Science, International Christian University, Osawa, Mitaka, Tokyo, 181-8585, Japan4Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan

5Institut fur Kernphysik (Theorie), Forschungzentrum Julich, D-52425 Julich, Germany6Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, DC 20052

7Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602(Received 25 October 2004; published 31 March 2005)

We analyze the problem of how to determine the parity of the + pentaquark in the reaction γ N → K →NKK, where N = n, p. Our model calculations indicate that the contribution of the nonresonant backgroundof the reaction γ N → NKK cannot be neglected and that suggestions to determine the parity based solely onthe initial-stage process γ N → K cannot be implemented cleanly. We discuss the various mechanisms thatcontribute to the background. Among the observables considered in this work, the triple spin asymmetry shows akinematical “window” where the predictions are sensitive to the + parity and insensitive to the various reactionmechanisms considered, and thus, this observable offers a possibility of determining the parity of the +.

DOI: 10.1103/PhysRevC.71.035203 PACS number(s): 13.60.Rj, 13.75.Jz, 13.85.Fb

I. INTRODUCTION

The first evidence for the + pentaquark discovered bythe Laser Electron Photon facility at SPring-8 (LEPS Collab-oration) [1] was subsequently confirmed in other experiments[2–7]. None of these experiments could determine the spin andparity of the +. Some proposals to do this in photoproductionprocesses using single- and double-polarization observableswere discussed in Refs. [8–10]. The difficulty in determiningthe spin and parity of + in the reaction γ n → +K− is dueto the way the pentaquark state is produced. The models of the+ photoproduction from the nucleon based on the effectiveLagrangian approach have been developed in Refs. [8–18]. Ashas been pointed out, there are great ambiguities in calculatingthe (spin) unpolarized and polarized observables. In the effec-tive Lagrangian formalism, the problems are summarized asfollows:

(1) Dependence on the coupling operator for the +NK

interaction, i.e., whether one chooses pseudoscalar (PS)or pseudovector (PV) couplings. In the case of PV cou-pling, gauge invariance requires a Kroll-Ruderman-typecontact term even for undressed particles which affectsboth unpolarized and polarized observables. For dressedparticles, in a tree-level description, contact currents alsoare required for PS coupling.

(2) Ambiguity due to the choice of the coupling constants. Atthe simplest level, five unknown coupling constants andtheir phases enter the formalism: gNK in the +NK

interaction, the vector and tensor couplings gNK∗ andκ∗, respectively, in the +NK∗ interaction, and the tensorcoupling κ in the electromagnetic γ+ interaction. Tofix the absolute value of gNK , one can use the relationbetween gNK and the decay width . This provides,however, only an upper limit for |gNK | because allthe experiments give only upper limits of the decay

width (about 25 MeV) which are comparable with theexperimental resolution.

(3) Dependence on the choice of the phenomenological formfactors: (i) form factors suppress the individual channelsin different ways, and (ii) form factors generate (modify)the contact terms for the PS (PV) coupling schemes whichaffect the theoretical predictions.

A possible solution to these problems is to use morecomplicated “model-independent” (triple) spin observables,discussed by Ejiri [19], Rekalo and Tomasi-Gustafsson [20],and Nakayama and Love [21]. These spin observables involvethe linear polarization of the incoming photon, and the polar-izations of the target nucleon and the outgoing +. Using basicprinciples, such as the invariance of the transition amplitudeunder rotation, parity inversion and, in particular, the reflectionsymmetry with respect to the scattering plane, one can arriveat unambiguous predictions which depend only on the +parity in the reaction γ N → +K . The key aspect of themodel-independent predictions is that in the final state the totalinternal parity of outgoing particles is different for positiveand negative + parities. We skip the discussions of thepractical implementation of using the triple spin observablessince experimental observations of the spin orientation of thestrongly decaying + are extremely difficult. Instead, wefocus on the basic aspects of this idea. For simplicity, in thefollowing, we limit our discussion for determining the +parity to isoscalar spin-1/2 +. In fact, most theories predictthe JP of + to be 1/2+ or 1/2−. For completeness, we notethat different aspects of the pentaquark properties in differentapproaches have been considered in Refs. [22–25]. For amore complete review, see a recent paper [26] and referencestherein.

There are two difficulties in applying the “model-independent” formalism for the parity determination. First, thefinal state in the photoproduction experiment is the three-body

0556-2813/2005/71(3)/035203(19)/$23.00 035203-1 ©2005 The American Physical Society

A. I. TITOV, H. EJIRI, H. HABERZETTL, AND K. NAKAYAMA PHYSICAL REVIEW C 71, 035203 (2005)

state NKK (and not the two-body +K) state. The spinobservables in the initial and final channels are deduced bytheir parities irrespective of the intermediate + parity. It isdifficult, therefore, to find the pertinent “model-independent”observables for this case. Second, the contribution of thenonresonant background of the reaction γ N → NKK cannotbe neglected. The observed ratio of the resonance peak tothe nonresonant continuum reported in the photoproductionexperiments is about 1.5−2. This means that the differencebetween the resonant and nonresonant amplitudes is not solarge and, therefore, the nonresonant background may modifythe spin-observables considerably.

The aim of the present paper is to discuss these importantaspects. We show that strict predictions for the γ N → +K

reactions lose their definiteness in the case of the γ N →NKK processes, where + decays strongly into NK; theybecome model dependent. Nevertheless, we try to identifythe kinematic regions where this dependence is weak, anda clear difference is expected for different + parities. Inthe following discussion, the term “resonant” is applied toprocesses which proceed through the virtual + state, whereasthe term “nonresonant” is used for all other processes. Thelatter may have intermediate nonexotic resonant states as well.The resonant amplitude consists of s-, u-, and t-channel termsand the contact (c) term defined by the +NK interaction,as depicted in Figs. 1(a)–(d). We also have a t-channel K∗exchange, as shown in Fig. 1(e). We found that the maincontribution to the “nonresonant” background comes from thevirtual vector-meson photoproduction γ N → V N → NKK,depicted in Figs. 2(a)–(c). We also have the excitation ofthe virtual scalar (σ ) and tensor (f2, a2) mesons shownin Figs. 2(d)–(f ), respectively, and we found that theircontribution in the near-threshold region with Eγ ≈ 2 GeVis negligible.

We write gNK∗ = αgNK in order to be able to pull out anoverall factor of g2

NK from all contributions shown in Fig. 1.In view of the proportionality g2

NK ∝ , the dependenceof the amplitudes on then disappears if we considerthe observables at the resonance position. The total amplitude

Θ+

KK

Θ+

K

Θ+

K

*K

Θ+

K

Θ+Θ+

(b)

N N

N

γK

(a)

γ

N N

K

K

NN

γK

(d)

γ

N N

K

(e)

(c)

NN

γK

FIG. 1. Tree-level diagrams for the reaction γ N → +K → NKK.

K

K

K

K

(a)

(d)

γ

N

K

φη Pπ

(b)

(e)

γ

NN

Kρ

(f)

γ

N

ω

γ

NN

KP π σ

ργ

N

ω

(c)

f2

γ

N

Kρ

σ a2

π

N

N

K

K

N

N

K

K

FIG. 2. Diagrams for the background process for the γ N →MN → NKK reaction, where M denotes the mesons ρ, ω, φ, σ, f2,and a2.

depends on the relative-strength parameter α which will befixed by experimental data. The dominance of the K∗ exchangechannel [16,17] allows us to reduce the number of relevantinput parameters at a given coupling scheme to four: the signand absolute value of α = gNK∗/gNK and the sign andabsolute value of κ∗.

We analyze two reactions, γp → pK0K0 and γ n →nK+K−, and we refer to them as the γp and the γ n reactions,respectively.

In Sec. II, we describe our formalism. In Sec. III, wediscuss the nonresonant background. The procedure to fix theparameters for the resonant amplitude is discussed in Sec. IV.The results of our numerical calculations for unpolarized andspin observables are presented in Sec. V. Our summary isgiven in Sec. VI. In Appendix A, we show an explicit formof the transition operators for the resonance amplitude. InAppendix B, we discuss the Pomeron exchange amplitude, andin Appendix C, we provide the parameters of the backgroundamplitude.

II. FORMALISM

A. Kinematics and cross sections

The scattering amplitude T of the γ N → NKK reaction isrelated to the S matrix by

Sf i = δf i − i(2π )4δ(4)(k + p − q − q − p′)Tf i, (1)

where k, p, q, q, and p′ denote the four-momenta of theincoming photon, the initial nucleon, the outgoing K and K

mesons, and the recoil nucleon, respectively. The standardMandelstam variables for the virtual + photoproduction aredefined by t = (q − k)2, s ≡ W 2 = (p + k)2. The K mesonproduction angle θ in the center-of-mass system (cms) isgiven by cos θ = k · q/|k||q|. The + decay distribution isdescribed by the polar () and azimuthal ( ) angles of theoutgoing kaon, with solid-angle element dZ = d cos d .In the center-of-mass system, the quantization axis (z) ischosen along the photon beam momentum, and the y axis isperpendicular to the production plane, i.e., y = k × q/|k × q|.The + decay distribution is analyzed in the + rest frame,where the quantization axis Z is chosen along the incoming(target) nucleon and Y = y. For completeness, the production

035203-2

SOME ASPECTS OF + PARITY DETERMINATION IN THE REACTION . . . PHYSICAL REVIEW C 71, 035203 (2005)

N

Θ+

K

Kz

xy

(b)

N

K

"D"

"P"

X

YZ

ΦΘ

(a)

γ

"P"

N

θγ

′

FIG. 3. Schematic description of the + production in (a) theγ N → K+ reaction in the center-of-mass system and (b) thereaction γ N → NKK in the + rest frame. The notations “P” and“D” correspond to the production and decay planes, respectively.

and decay planes with the corresponding coordinate systemsare depicted in Fig. 3. We use the convention of Bjorken andDrell [27] to define the γ matrices; the Dirac spinors arenormalized as u(p)γαu(p) = 2pα . The invariant amplitude Tf i

is related to Tf i by

Tf i = Tf i√(2π )15 2|k| 2EK (q) 2EK (q′) 2EN (p) 2EN (p′)

,

(2)

where Ei(p) =√

M2i + p2, with Mi denoting the mass of

particle i. The differential cross section is related to theinvariant amplitude by

dσf i

ddZdM

=pF

√λ(s,M2

,M2K

)32(2π )5W

(W 2 − M2

N

) 1

4

∑mi,mf ,λγ

|Tf i |2,

(3)where λ(x, y, z) is the standard triangle kinematics function;

pF =√

λ(M2,M2

N,M2K )/2M is the + decay momentum;

M is the invariant mass of the outgoing nucleon and K meson;mi and mf are the nucleon spin projections in the initial andthe final states, respectively; and λγ is the incoming photonhelicity.

In the following, we consider the observables at theresonance position where the invariant mass of the outgoingnucleon and K meson is equal to the + mass, M = M0 =1540 MeV. In this case, the invariant amplitude of the resonantpart is expressed through the + photoproduction (A) and the+ decay (D) amplitudes according to

T ±mf ;mi,λγ

=∑m

A±m;mi,λγ

1

M

D±m;mf

, (4)

where plus (minus) corresponds to the positive (negative) +

parity (JP = 12

±), and m is the + spin projection. In the

+ rest frame, the decay amplitudes (p and s waves for thepositive and negative + parity) read

1

M

D+m;mf

= D0(2m δm,mf

cos + δ−m,mfe2im sin

),

1

M

D−m;mf

= −D0 δm,mf, D0 =

√4π

pF

, (5)

where is the total width of the + decay. After integratingover the decay angles (dZ) in Eq. (3), one obtains

dσRf i

d dM

∣∣∣∣∣M=M0

= 1

π

dσ+f i

d, (6)

where

dσ+f i

d=

√λ(s,M2

,M2K

)64π2W 2

(W 2 − M2

N

) 1

4

∑mi,mf ,λγ

∣∣Amf ;mi,λγ

∣∣2

(7)is the cross section of the + photoproduction in the γ N →+K reaction, with mf = m. By using the linear relationgNK∗ = αgNK , one finds that dσR/ddM does notdepend on the + decay width at the resonance position,whereas dσ+

/d does.

B. Effective Lagrangians for the resonant amplitudes

As mentioned above, we describe the basic resonanceprocess by considering the photoproduction of +, with asubsequent decay of + into a nucleon and a kaon, as shownin Figs. 1(a)–(d). We neglect, therefore, the contributionsresulting from the photon interacting with the final decayvertex (see Ref. [8] for the corresponding three additionalgraphs). In view of the chosen kinematics, where the invariantmass of the final KN pair is at the resonance position, this isa good approximation since in the neglected graphs the +is far off-shell and the graphs of Figs. 1(a)–(d) dominate theresonance contribution. From a formal point of view, then, welose gauge invariance of the process since this necessarily alsorequires the contributions arising from the electromagneticinteraction with the decay vertex. However, following Ref. [28]for the initial photoproduction process, we construct an overallconserved current by an appropriate choice of the contact termof Fig. 1(d). In view of the dominance of the resonance graphswe do take into account, numerically we expect very littledifference between our present current-conserving results andthose of a full gauge-invariant calculation.

The effective Lagrangians which define the amplitudesshown in Figs. 1(a)–(d) are discussed in Refs. [8–16].Note, different papers employ different phase conventions.Therefore, for easy reference, we list here the explicit forms ofthe effective Lagrangians used in the present work. We have1

LγKK = ie (K−∂µK+ − K+∂µK−)Aµ, (8a)

Lγ = −e

(γµ − κ

2M

σµν∂ν

)Aµ, (8b)

LγNN = −e N

(eNγµ − κN

2MN

σµν∂ν

)AµN, (8c)

1Throughout this paper, isospin operators will be suppressed in allthe Lagrangians and matrix elements for simplicity. They can beeasily accounted for in the corresponding coupling constants.

035203-3


L±[pv]NK = ∓ gNK

M ± MN

±µ (∂µK)N + h.c., (8d)

L[pv]γNK = −i

egNK

M ± MN

±µ AµKN + h.c., (8e)

L±[ps]NK = −igNK±N + h.c., (8f)

where A, ,K , and N are the photon, the +, the kaon, andthe nucleon fields, respectively; ±

µ ≡ ±γµ (with + = γ5

and − = 1 for positive and negative parity, respectively);ep = 1; en = 0; and κN is the nucleon anomalous magneticmoment (κp = 1.79 and κn = −1.91). Equation (8e) describesthe contact (Kroll-Ruderman) interaction in the pseudovectorcoupling scheme [see Fig. 1(d)], which is absent in the case ofthe pseudoscalar coupling [Eq. (8f)]. In addition, we considerthe K∗ exchange process shown in Fig. 1(e), which is describedby the two effective Lagrangians

LγKK∗ = egγKK∗

MK∗εαβµν∂αAβ∂µK∗

ν K + h.c., (9a)

L±NK∗ = −gNK∗ ∓

(γµ − κ∗

M + MN

σµν∂ν

)×K∗µN + h.c. (9b)

In calculating the invariant amplitudes, we dress the verticesby form factors. In the present tree-level approach, withour chosen kinematics, only the lines connecting the elec-tromagnetic vertex to the initial +KN vertex correspondto off-shell hadrons. We describe the product of both theelectromagnetic and the hadronic form-factor contributionsalong these off-shell lines by the covariant phenomenologicalfunction

F (M,p2) = 4

4 + (p2 − M2)2, (10)

where p is the corresponding off-shell four-momentum, M isthe mass, and is the cutoff parameter. We conserve theelectromagnetic current of the entire amplitude by making theinitial photoproduction process gauge invariant. To this end,we apply the gauge-invariance prescription of Haberzettl [28],in the modification by Davidson and Workman [29] (whichrenders the required additional contact terms free of kinemati-cal singularities), to construct a contact term for the initial pro-cess γ N → +K . For pseudovector coupling, the inclusionof form factors not only modifies the usual Kroll-Rudermanterm but also requires additional contact terms contributing tothe amplitudes. We emphasize that contributions of the lattertype are necessary even for pure pseudoscalar coupling.

The resonance amplitudes obtained for the γ n and γp

reactions read

A±f i(γ n) = u(p)

[Ms±

µ + Mt±µ + Mu±

µ + Mc±µ

+Mt±µ (K∗)

]up(p)εµ, (11a)

A±f i(γp) = u(p)

[Ms±

µ + Mu±µ + Mc±

µ

+Mt±µ (K∗)

]up(p)εµ. (11b)

The explicit forms of the transition operators Miµ for

the γ n → +K− → nK−K+ and γp → +K0 → pK0K0

reactions are shown in Appendix A. The choice of the strengthparameters gi in the effective Lagrangians, and the cutoffparameters = B , is discussed in Sec. IV.

III. NONRESONANT BACKGROUND

The nonresonant background for the reaction γ n →nK+K− has been discussed qualitatively by Nakayamaand Tsushima [8]. Together with the virtual vector-mesoncontribution V → K+K− (V = ω, ρ, and φ mesons), theyhave included the excitation of the virtual hyperons. Thecoupling constants in the corresponding effective Lagrangianswere fixed using the known decay widths and SU(3) symmetry,and the particles were taken as undressed but with physicalmasses. The present analysis with dressing form factors showsthat at Eγ ≈ 2 GeV, the main contribution comes from thevirtual vector-meson photoproduction. For completeness, wealso explore the contribution from the scalar (σ ) and tensor(a2, f2) mesons. As we show in Sec. III E, the latter is foundto be negligibly small in the Eγ ≈ 2 GeV region.

A. Vector meson contributions: ρ,ω, φ

Naively, one can expect the dominance of the intermediateρ-meson channel in the background contribution because thecross section for the real ρ-meson photoproduction is about anorder of magnitude larger than that for the ω meson, and it isabout two orders of magnitude larger than that for the φ-mesonphotoproduction. However, at Eγ ≈ 2 GeV, the KK invariantmass is distributed in the region 1 GeV <∼ MKK <∼ 1.2 GeV,which straddles the φ mass. Figure 4 shows an example ofthe phase-space diagram of the KK invariant mass versus thecosine of the K decay angle at a fixed K production angle ofθ (cm) = 55. One can see that the narrow mass distribution ofthe φ is within the sampled kinematic region, which thereforemakes the φ-meson contribution significant.

In this study, we consider the contribution from ρ, ω, and φ

mesons. The low-energy ρ- and ω-meson photoproductions aredescribed within an effective meson-exchange model [30–32].

−1.0 −0.5 0.0 0.5 1.0

cosΘ

0.9

1.0

1.1

1.2

1.3

MK

K

θ=55o

FIG. 4. The phase-space diagram for the + photoproduction atEγ = 2 GeV: the invariant mass MKK versus cos at a fixed K

production angle (θ = 55). The cutoff area of the phase phase isplaced between the two solid lines.

035203-4


Thus, the ρ-meson photoproduction is dominated by thet-channel scalar (σ ) and pseudoscalar (π ) meson exchanges.The ω-meson photoproduction is mostly defined by the π

exchange. In Ref. [33], the σ exchange in the ρ photopro-duction is reexamined on the basis of uncorrelated two-pionand tensor f2-meson exchanges. The main problem of thisapproach is the requirement of a large coupling constant for thef2ργ interaction. This results in a branching ratio of f2 → ργ

decay that seems to be unreasonably large. Another ambiguityis related to the unknown f2NN coupling and the form factorsfor the off-shell f2 meson. Since the quality of the descriptionof the experimental data using either the σ exchange or thef2 exchange is comparable [33], and to avoid introducing toomany unknown parameters, we employ the σ -exchange modelin this work; this is quite reasonable for the present purposes.When the photon energy increases, we have to add the Pomeronexchange as well. But at Eγ ≈ 2 GeV, it is important only forthe ρ channel where the Pomeron exchange gives about 30%of its contribution. In the ω channel, the Pomeron contributionis suppressed by the factor (γρ/γω)2 ≈ 6.33/72.71 ≈ 1/11.5,where γρ and γω are the ρ and ω decay couplings, respectively.Therefore, in the ω photoproduction, we limit ourselves tothe π exchange process only. The effective Lagrangiansresponsible for the meson-exchange channels read

LπNN = gπNN

MN

Nγ5γµ∂µπN, (12a)

LσNN = gσNNNNσ, (12b)

Lρσγ = egρσγ

2Mρ

(∂νρµ − ∂µρν)(∂νAµ − ∂µAν)σ, (12c)

LV πγ = egV πγ

MV

εαβµν∂αAβ∂µVνπ, (12d)

LV KK = igV KK (K∂µK − K∂µK)Vµ, (12e)

where V stands for the vector meson. The amplitudes for theγ N → NV → NKK reaction may be expressed as

Tf i =∑λV

AVmf λV ;miλγ

1

M2V − M2

KK+ iV MV

DVλV

FV

(M2

KK

),

(13)

where AV and DV are the vector-meson photoproduction(γ N → VN) and decay (V → KK) amplitudes, respectively;

MKK is the KK invariant mass; V is the total decay width ofthe vector meson; λγ and λV are the helicities of the photonand vector meson, respectively; and FV is the form factor ofthe virtual vector meson.

The photoproduction amplitudes may be expressed in astandard form

AVmf λV ;miλγ

= ufMµνui εµλγ

ε∗νλV

. (14)

In the case of the scalar (S: σ ) and pseudoscalar (PS: π ) mesonexchange, the transition operators Mµν read

MSµν = egρσγ gσNN

Mρ

gµν(k · QV ) − QV µkν

t − M2σ

Fσ (t ), (15)

MPSµν = i

egρπγ gπNN

Mρ

γ5εµναβkαQV β

t − M2π

Fπ (t ), (16)

respectively, where QV = q + q, and FM is the product of thetwo form factors of the virtual exchanged mesons in the MNNand γV M vertices. The explicit forms of FM (t ) are given inAppendix C. Regarding the four-momentum transfers to theKK pair, note that t is different from t in + photoproduction.

The decay amplitude,

DVλV

= gV KK (q − q)µεµλV

, (17)

has a simple form in the V rest frame, with the quantization axisz parallel to the beam momentum (Gottfried-Jackson system),i.e.,

DVλV

= 2kf gV KK

√4π

3Y1λV

(q), kf = MKK

2

√1 − 4

M2K

M2KK

,

(18)

where q is the solid angle of the direction of flight of the K me-son in the KK rest frame, i.e., q ≡ K . The φ-meson photo-production is defined by the Pomeron and pseudoscalar (π, η)meson exchanges, as described in Ref. [34]. For easy reference,we provide the transition operator MP

µν for the Pomeronexchange amplitude, and the parameters which define theφ-, ρ-, and ω-meson photoproduction, in Appendixes Band C, respectively.

Figures 5(a)–(c) show the cross sections of the vector-meson photoproduction in the reactions γp → Vp (V =ρ, ω, φ) at Eγ ≈ 2 GeV together with the available data. One

0.0 0.5 1.0 1.5

−t [GeV2]

100

101

102

dσ/d

t [µb

/GeV

2 ]

dσ/d

t [µb

/GeV

2 ]

dσ/d

t [µb

/GeV

2 ] Eγ=1.92 GeV

γp >ρp

100

10-1

101

102

0.0 0.4 0.8 1.2

−t [GeV2]

Eγ=1.92 GeV

γp >ωp

10-2

10-1

102

0.0 0.4 0.8 1.2

−t [GeV2]

Bonn

SAPHIR

γp >φp

Eγ=2 GeV

(a) (b) (c)

– ––

FIG. 5. Cross sections for the ρ-, ω-, and φ-meson photoproduction. The experimental points are taken from Refs. [35–37].

035203-5


can see quite a reasonable agreement between the experimentaldata and the calculation. This encourages us to use the samemodel for the description of the nonresonant background inthe + photoproduction.

B. Scalar meson contribution: σ

The γρσ interaction responsible for the ρ-meson contribu-tion naturally leads to the virtual σ -meson photoproductionand its subsequent decay into the KK pair as shown inFig. 2(d). The corresponding effective Lagrangians for thistransition read

LσKK = MσgσKKσKK, (19a)

LρNN = −gρNN

(NγµρµN − κρ

2MN

Nσµν∂νρµN

). (19b)

The σKK coupling may be estimated from SU(3) symmetryas gσKK = −gσππ/2.

The amplitude for the γ N → σN → NKK process has theform

Amf ;miλγ= ufMρ

µuiεµλγ

, (20)

where

Mρµ = egρσγ gρNN

Mρ

(t − M2

ρ

) [(1 + κρ)Rµ − κρ

R′µ

MN

]Fρ(t ), (21)

with

Rµ = (k · Q)γµ − k/Qµ, (22a)

R′µ = (k · Q)p′

µ − (k · p′)Qµ, (22b)

Q = p′ − p. (22c)

The σ → KK decay amplitude is a constant

Dσ = −MσgσKK

√4πY00(q). (23)

For the vector and tensor coupling constants and the cutoffparameter in Fρ , we take the corresponding values from theBonn model [38] (see Appendix C).

C. Tensor meson contributions: a2, f2

The tensor a2(1320) and f2(1270) mesons have finitebranching ratios into the KK decay channel, and therefore,one can expect their nonnegligible contributions to thenonresonant background in the + photoproduction [39]. Thecorresponding branching ratios are (4.9 ± 0.8) × 10−2 and(4.6 ± 0.5) × 10−2 for the a2(1320) and f2(1270) mesons,respectively [41].

As with the scalar meson, the tensor mesons (a2 and f2)appear in the “nonresonant” background in two ways. First,a2 and f2 give a contribution to the photoproduction of theω and ρ mesons, respectively. Second, they can be produceddirectly by the incoming photon with subsequent exchange byω and ρ mesons, as depicted in Figs. 2(e) and (f ). The first caseresults only in some renormalization of the coupling constantsin the ω and ρ photoproduction amplitudes, and it does not

modify the shape of the KK invariant mass distribution. Butsince the coupling constants in the ρ and ω photoproductionprocesses are fixed by the corresponding experimental dataanyway, we may assume that such a renormalization effect istaken into account in those effective strength parameters. Thesecond contribution may change the KK invariant distributionqualitatively when MKK ≈ Mf2(a2). This is realized at higherenergies with Eγ 2.3 GeV. At Eγ ≈ 2 GeV, their contri-bution is not very strong; nevertheless, for completeness, weinclude these processes in our consideration. We assume thatproduction of the a2 and f2 mesons is generated by the a2γω

and f2γρ interactions, respectively.The interaction of the tensor and the two vector fields is

described by the gauge-invariant interaction

Lt2V1V2 = gt2V1V2

Mt2

(Lαβ + Lβα)ξαβ,

(24)Lαβ = (

∂αVµ

1 − ∂µV α1

)(∂βV2µ − ∂µV

β

2

),

where V1,2 and ξ are the vector and tensor meson fields,respectively, and t2 = a2, f2.

The t2KK interaction is described by the effectiveLagrangian

Lt2KK = gt2KK

Mt2

(∂βK∂αK + ∂βK∂αK)ξαβ. (25)

The coupling constant gt2KK is related to the decay widtht2→KK that subsumes transitions into both K0K0 and K+K−channels according to

g2t2KK = 15πt2→KKM4

t2

2p5t2

, (26)

where pt2 = Mt2

√1/4 − M2

K/M2t2 is the t2 decay momentum.

Here, we assume that t2→K+K− ≈ t2→K0K0 ≈ 0.5t2→KK .Using the known widths a2→KK and f2→KK from [41], weget

ga2KK = 4.9, gf2KK = 7.4. (27)

The amplitude of the γ N → Nt2 → NKK transition isexpressed as

Tf i =∑λγ σ

At2mf σ ;miλγ

1

M2t2 − M2

KK+ it2Mt2

Dt2σ Ft2

(M2

KK

),

(28)where At2 is the tensor meson photoproduction (γ N → t2N )amplitude, t2 is the total decay width of the tensor meson,σ is the spin projection of the tensor meson, and Ft2 is theform factor of the virtual tensor meson. The t2 → KK decayamplitude reads

Dt2σ = −2gt2KKk2

f

Mt2

√8π

15Y2σ (q). (29)

The a2-meson photoproduction amplitude is given by

Aa2mf σ ;miλγ

= ga2γωgωNN

Ma2

uf Ha2αβµ ui ε

∗αβ εµλγ

Fω(t ), (30)

035203-6


0.9 1.0 1.1 1.2 1.3

MKK (GeV)

10-1

100

101

102

dσ/d

tdM

KK (

µb/G

eV3 )

2.00.90.6

Λ0 GeV(γn)

(a)

(b)φ

Invariant K +K - mass (GeV/c2) E

vent

s/(0

.005

GeV

/c2 )

1

10

102

1.0 1.1 1.2 1.3

FIG. 6. (a) Dependence of the shape ofthe invariant mass distribution on the cutoffparameter 0 in the γ n → K+K− reaction.(b) KK invariant mass distribution in γ n →K+K− taken from Ref. [1]; arrows indicate theφ-meson cut window.

where εαβ is the Rarita-Schwinger spinor of the tensor meson

εαβ (σ ) =∑l1l2

〈1l1 1l2|2σ 〉 εα(l1)εβ(l2), (31)

and

Ha2αβµ = (Qαγ ν − Qνγα)(kβgµν − kνgµβ)

+ (Qβγ ν − Qνγβ)(kαgµν − kνgµα). (32)

Similarly, the amplitude of the f2-meson photoproductionreads

Af2mf σ ;miλγ

= gf2γρgρNN

Mf2

uf Hf2αβµ ui ε

∗αβ εµλγ

Fρ(t ), (33)

with

Hf2αβµ = (QαGν − QνGα)(kβgµν − kνgµβ)

+ (QβGν − QνGβ)(kαgµν − kνgµα), (34)

Gµ = (1 + κρ)γµ − κρ

MN

p′µ.

In the absence of experimental information necessary toextract the coupling constants ga2γω and gf2γρ , we assume that

ega2ωγ = 0.29, egf2ργ = 0.14. (35)

These are rough estimates obtained by making use of theavailable data for the decay widths for a2 → ωπ+π−, a2 →γ γ , and f2 → γ γ in conjunction with the vector dominancemodel, in addition to the Gell-Mann–Sharp–Wagner contactterm [40].2

2The value of the coupling constant ga2ωγ used in the present workleads to the a2 → ωγ decay width

a2→ωγ ≈ 0.29 MeV,a2→ωγ

tot≈ 2.76 × 10−3,

which is comparable to the a2 → πγ decay with the branching ratio(2.68 ± 0.31) × 10−3. Our estimate, therefore, may be taken as anupper limit. The value of the coupling constant gf2ωγ in Eq. (35)results in the branching ratio

f2→ργ ≈ 0.059 MeV,f2→ργ

tot≈ 3.2 × 10−4,

which is about a factor of 28 greater than the branching ratiofor the f2 → γ γ decay, which is (1.14 ± 0.13) × 10−5. Note thatour estimate is about a factor of 5.5 smaller than the similar estimategiven in Ref. [33], where f2→ργ /f2→γ γ ≈ 155.

D. K K invariant mass distribution

If there is no limitation on the phase space of the outgoingkaons, then the integration over the decay angle of the K mesoneliminates the interference terms among the scalar, vector, andtensor meson exchange contributions, and the KK invariantmass distribution may be expressed in a compact form as

64π3(s − M2

N

)2

kf

dσ

dt dMKK

= M2σ g2

σKK

|Aσ |2(M2

KK− M2

σ

)2 + M2σ2

σ

+ 4k2f

3

∣∣∣∣∣∑V

gV KKAVf i

M2KK

− M2V + iMV V

∣∣∣∣∣2

+ 8k2f

5

∣∣∣∣∣∑t2

kf

Mt2

gt2KKAt2f i

M2KK

− M2t2 + iMt2t2

∣∣∣∣∣2

, (36)

where the average over the spins/helicity in the initial stateand the summation over the spin variable in the final statesare implied. One can see that, near threshold, the tensor mesoncontribution is suppressed by the factor k2

f /M2t2

1 comparedto the vector meson contribution.

The KK invariant mass distribution at a forward angle ofKK photoproduction [θKK (cm) = 10] is shown in Figs. 6and 7. The photon energy was taken from the threshold to2.35 GeV, in accordance with the measurement of Ref. [1]. Thedistribution has one unknown parameter compared to the realvector meson photoproduction. It is the cutoff parameter 0

in the form factor F(V,σ,a2,f2) = F (M(V,σ,a2,f2),M2KK

), whereF (M,x2) is defined in Eq. (10). The dependence of theKK invariant mass on this parameter is shown in Fig. 6(a).A comparison with the measured distribution [1] shown inFig. 6(b) favors 0 ≈ 0.9 GeV. We use this value in our furtheranalysis. Figures 7(a) and (b) show the structure of thesedistributions. One can see a strong φ-meson photoproductionpeak at MKK ≈ Mφ and a long tail dominated by the ρ-mesonchannel. The contribution from the other mesons is muchsmaller. The contribution from the a2 and f2 mesons becomescomparable to that from the vector mesons near the threshold,MKK ≈ 1.3 GeV (at Eγ = 2.2 GeV). In this region of the KK

invariant masses, the cross section is rather small compared to

035203-7


0.9 1.0 1.1 1.2 1.3

MKK (GeV)

10-2

100

102

dσ/

dtdM

KK (

µb/G

eV3 )

φ

ω

f2

ρ

(γn)

σa2

(a)

0.9 1.0 1.1 1.2 1.3

MKK (GeV)

10-2

100

102

dσ/

dtdM

KK (

µb/G

eV3 )

φ

ωf2

ρ

(γp)

σa2

(b)

FIG. 7. KK invariant-mass distribution in(a) γ n → K+K− and in (b) γp → pK0K0. Thearrows indicate the φ-meson cut window.

that around MKK ≈ Mφ , which gives the main contribution tothe background of the + photoproduction at Eγ ≈ 2 GeV.

Finally, we note that the cross section for γ n → nK+K−exceeds that for γp → pK0K0 by approximately a factor oftwo. This is due to the distinct φ − ρ interference effect in thesetwo reactions caused by the different signs in the ρK+K− andρK0K0 coupling constants. In the case of the φ meson, thesigns of the φK+K− and φK0K0 coupling constants are thesame.

E. Nonresonant background in + photoproduction

Let us now examine the background contribution to theangular distribution of the KK pair in the final state. TheKK invariant mass is taken at the + resonance position atM = M0 = 1.54 GeV. The corresponding differential crosssection, dσ/ddM, where is the solid angle of theKK pair, is calculated using the general expression for thedifferential cross section given by Eq. (3). Here, all the back-ground channels contribute coherently, and the integration overthe + decay angle Z is performed numerically.

Our result for the γ n → nK+K− reaction is shown inFig. 8(a). One can see a rather strong contribution from theφ photoproduction. We will eliminate the phase space withthe KK invariant mass from 1.00 to 1.04 GeV followingRef. [1] in order to reduce its contribution. The correspondingphase space, shown schematically in Fig. 4, is about 15%of the total phase space, but it gives about 100% of theφ-channel contribution to the background. The differentialcross sections, with an MKK cut for the γ n and γp reactions,are shown in Figs. 8(b) and 8(c), respectively. Now the φ

and ρ contributions are comparable to each other. Anotherinteresting aspect is the enhancement of the σ contributioncompared to the corresponding contribution in the case of theKK invariant mass distributions shown in Figs. 6(a) and (b).This enhancement is explained by the term κρ(R′ · εγ )/MN inEq. (20) caused by the ρNN tensor coupling. The contributionfrom this term to the differential cross section increasesstrongly with increasing |p′| sin θK . When the momentumtransfer to the KK pair is small [see Figs. 7(a) and (b)], thiscontribution is rather small, whereas in the kinematic conditionof Fig. 8 it is large.

As can be seen from Fig. 8 and from the discussionin connection with Figs. 7(a) and (b) in Sec. III D, thecontributions from the tensor mesons, a2 and f2, at Eγ ≈2 GeV are found to be very small (even for large couplingconstants [33] and different choices of their signs). Therefore,hereafter, the tensor mesons will be omitted in our calculations.

IV. FIXING THE PARAMETERS OF THERESONANT AMPLITUDE

(1) The magnitude of the coupling constant, gγKK∗ , isextracted from the width of the K∗ → γK decay [41].Its sign is fixed by SU(3) symmetry. We have egγK0K∗0 =−0.35 and egγK±K∗± = 0.23.

(2) The contribution of the s channel [Fig. 1(b)] is small,which leads to a rather weak dependence of the totalamplitude on the tensor coupling κ in the γ

interaction within a “reasonable” range of 0 <∼ |κ| <∼0.5 [42]. Therefore, we choose κ = 0 for both parities.

0 135 180

θ (degrees)

10-4

10-3

10-2

10-1

10-4

10-3

10-2

10-1

10-4

10-3

10-2

10-1

dσ/

dΩdM

Θ (

µb/s

r G

eV)

tot

no cut

ρφ

ωσ

(γn)(a)

a2

f2

0 90 135 180

θ (degrees)

dσ/

dΩ

dMΘ

(µb

/sr

GeV

)

tot

with cut

ρ φω

σ

(γn)

(b)

a2

f2

900 45 135 180

θ (degrees)

dσ/

dΩ

dMΘ

(µb

/sr

GeV

)

tot

with cut

ρ φω

σ

(γp)(c)

a2

f2

9045 45

FIG. 8. Contribution of the background processes to the differential cross section of the + photoproduction for the (a, b) γ n and (c) γp

reactions (a) without and (b, c) with the cut in the KK invariant mass distribution, respectively.

035203-8


0 135 180

θ (degrees)

10-5

10-3

10-1

dσ/

dΩ

dMΘ

(µb

/sr

GeV

)

t (K*–exch.)

BG

1/2+(γp)

s+u+c

PSPV

(a)

0 45 90 135 180

θ (degrees)

10-5

10-4

10-3

10-2

10-1

dσ/

dΩ

dMΘ

(µb

/sr

GeV

)

t (K *–exch.)

BG

1/2

s+u+cPSPV

(b) (γp)

0 45 90 135 180

θ (degrees)

10-5

10-4

10-3

10-2

10-1

10-5

10-4

10-3

10-2

10-1

dσ/

dΩ

dMΘ

(µb

/sr

GeV

)

t (K *–exch.)BG

1/2+ -

-

(γn)

s+t+u+cPS

PV

(c)

0 45 90 135 180

θ (degrees)

dσ/d

ΩdM

Θ (

µb/s

r G

eV)

t (K *–exch.) BG

1/2(γn)

s+t+u+c

PSPV

(d)

9045 FIG. 9. Contributions of the Born s, t, u and thecontact c terms in the (a, b) γp and (c, d) γ n reactions,together with the t-channel K∗ exchange and thebackground processes. The respective cases for positiveand negative π are depicted in (a, c) and (b, d).

(3) The coupling constant gNK for the positive and negative+ parity is found from the + decay width,

=[g±

NK

]2pF

2πM

(√M2

N + p2F ∓ MN

). (37)

There are several indications that the + decay widthis most likely to be of the order of one MeV [43] andthat the observed width exp in the photoproductionis rather to be regarded as the experimental resolution(). By construction (see Sec. I), the differential crosssection for the + photoproduction defined by Eq. (3) atthe resonance position does not depend on the decaywidth. Instead, it depends on the ratio of the couplingconstants gNK∗ and gNK (as well as, in general, otherparameters). In principle, this ratio may be extractedfrom the experimental data by comparing the resonantand background contributions because the cross sectiondue to the background is known in the present case.However, a proper comparison requires accounting forthe “experimental resolution” in our calculation. Thesmearing of the + invariant mass distribution results in asuppression of the resonant cross section at the resonanceposition by a factor of

d2 ≈

≈ 1

10, (38)

where we use an averaged value of ≈ 20 MeV and ≈ 2 MeV. We include the effect of this smearing bymultiplying all the resonance amplitudes by this factor d.In our calculation, we use the ratio of the resonance plusbackground to background processes from the experimentto fix the model parameters. This ratio is different indifferent experiments and varies from 2 [3] to 7 [5]. Wewill use a value of 3.4, which corresponds to that foundin the LEPS experiment.

(4) The next important point is to fix the cutoff parameters. Inour model, we have two different cutoff parameters. One(K∗ ) is in the t-channel K∗ exchange amplitude. Anotherone (B) defines the Born terms of the s, u, and t channelsand the current-conserving contact terms c. Generallyspeaking, K∗ together with gNK∗ (or the ratio α =gNK∗/gNK ) defines the strength of the K∗ exchangeamplitude. Increasing K∗ leads to a decreasing α. Ifwe assume for the moment that the Born terms arenegligible compared to the K∗ exchange, then taking asa guide the quark model prediction for α for the positive+ parity [44],

α ≈√

3, (39)

we can fix K∗ using the calculated background crosssection and measured ratio of the resonance contributionto the background [signal to noise (S/N )]. The γp →pK0K0 reaction is close to this ideal case, where theK∗ exchange contribution is much larger than the Bornterms. But in the case of the γ n → nK+K− reaction,the situation is more complicated. The t channels and thecontact terms are strong and comparable to the K∗ ex-change. These situations for the γp and γ n reactions maybe understood from Fig. 9. We show here the differentialcross sections for the γ N → NKK reactions as a functionof K-production angle in the center-of-mass system atthe resonance position and for positive and negative +parities. We display the individual contributions from theBorn terms, the K∗ exchange channel, and the backgroundprocesses for both the pseudoscalar and pseudovectorcouplings with B = 1 GeV. The parameters for the K∗exchange amplitude read ∗

K = 1.5, α = 1, and κ∗ = 0.One can see that for the γp reactions the K∗ exchangechannel is dominant, whereas for the γ n reactions, all

035203-9


0.4 0.8 1.2

ΛB (GeV)

0

2

4

6

α

1.0 PV

ΛK * (γn)PS

(a)

inf

1.5

0.6 0.9 1.2

ΛB (GeV)

0

2

4

6

α 1.0

PVΛK *

(γp)

PS

inf

1.5

(b)

FIG. 10. The scale parameter α as a function of thecutoff parameters K∗ and B .

the individual contributions become comparable to eachother, and the problem of fixing the cutoff parameter B

must be solved consistently. Note that the inclusion ofthe and photoproduction [45] processes results in alarger ambiguity in the choice of B , which varies from0.5 to 2 (GeV) depending on the coupling scheme, methodof conserving the electromagnetic current, etc.

To choose B , we use the following strategy. Weassume that the ratio α, as well as the cutoff K∗ , must bethe same in the γp and γ n reaction. Then, fixing α andK∗ from the γp reaction and using the signal-to-noiseratio S/N , we determine B unambiguously for the γ n

reaction, and its value will depend on the type of thecoupling (PV or PS), the value and the sign of tensorcoupling (κ∗), and the sign of α.

Figure 10 shows the dependence of α on B at fixed K∗

and κ∗ = 0. For the γp reaction, this dependence is ratherweak, but for γ n it is strong. For further analysis of the γp

reaction, we chose an averaged value of B = 1 GeV. Weanalyze observables at three values of the tensor coupling κ∗ =0,±0.5. The corresponding values of α depending on κ∗ andthe + parity are shown in Table I. All the calculations are doneat the K-production angle of θ = 55, where the resonanceprocesses are largest. Also, at this condition, the backgroundcontribution is dominated by the known φ and ρ channels,and therefore it is better established. Figures 11(a)–(d) showthe result for the differential cross section for the γ N → NKKreactions, calculated by considering a coherent sum of all theresonance processes with the K∗ exchange amplitude scaledby a factor of α and with an appropriate B . One can see thatin the vicinity of θ ≈ 55, the unpolarized cross sections forthe different coupling schemes and different parities are closeto each other for both the γp and γ n reaction. The differencebetween these two reactions may appear in the angular distri-bution of the + → NK decay and in the corresponding spinobservables.

TABLE I. Ratio α = gNK∗/gNK for different+ parity and tensor coupling κ∗ = 0, ±0.5.

1/2P \κ∗ 0.5 0.0 −0.5

1/2+ 1.67 1.875 2.011/2− 9.38 8.625 7.88

V. RESULTS AND DISCUSSION

A. + decay distribution

The angular distribution of the + → NK decay isdescribed by the decay amplitudes D± in Eqs. (4) and (5).Later we discuss the polar angle () distribution integratedover the azimuthal angle ( ). The decay amplitudes exhibita p- or an s-wave distribution depending on the parity of +(π) being positive or negative, respectively. However, if thespin state of the recoil nucleon is not fixed, the difference inthe angular distribution, normalized as∫

W (cos ) d cos = 1, (40)

disappears (for a spin- 12+). The pure resonant amplitude

results in an isotropic distribution with

WR(cos ) = 12 . (41)

The interference between the resonant and background am-plitudes leads to an anisotropy in the angular distribution.Figures 12(a) and (b) show the angular distribution W (cos )for the γ n and γp reactions. Here, we chose the PV-couplingscheme with positive α and κ∗ = 0. The results for other inputparameters are similar to those shown in Fig. 12. The solidand dashed curves correspond to the positive and negativeparities for +, respectively. The angular distribution dueto the background is shown by the dot-dashed curve, whereasthe contribution from the pure resonance channel is shown bythe solid thin line. Some nonmonotonic behavior of W (cos )around ≈ π/3 is caused by the sharp cut in the KK invariantmass with |MKK − Mφ| < 20 MeV as described in Sec. III E.We do not see any essential difference between the calculationscorresponding to different π.

The differential cross section due to the backgroundchannels in γp → pK0K0 is shown in Fig. 12(c). It ispeaked at cos ≈ −1 for the following two reasons. First,the momentum transfers to the KK pair (described by |t |)reaches its minimum value at cos = −1, and second, thepolar angle of the KK decay distribution (θK ) with respectto the photon momentum in the KK rest frame is closeto π/2 (when θ ≈ 55 and ≈ π ). In this region, thedominant background contribution arising from the vectormeson channels has a maximum because the cross sectiondue to the γ N → NV → NKK process is proportional tosin2 θK .

035203-10


0 45 90 135 180

θ (degrees)

10-3

10-2

10-1

100

10-3

10-2

10-1

100

10-3

10-2

10-1

100

10-3

10-2

10-1

100

dσ/d

ΩdM

Θ (

µb/s

r G

eV)

BG

1/2+(γp)

s+u+cPSPV

(a)

s+u+c+K*

0 45 90 135 180

θ (degrees)dσ

/dΩ

dMΘ

(µb

/sr

GeV

)

BG

1/2(γp)

s+u+cPSPV

(b)

s+u+c+K*

0 45 90 135 180

θ (degrees)

dσ/d

ΩdM

Θ (

µb/s

r G

eV)

BG

1/2+ -

-

(γn)

s+u+cPSPV

(c)

s+u+c+K*

0 45 90 135 180

θ (degrees)

dσ/d

ΩdM

Θ (

µb/s

r G

eV)

BG

1/2(γn)

s+u+cPSPV

(d)

s+u+c+K*

FIG. 11. The cross section of the resonance +

photoproduction and the background processes as afunction of the K-production angle for the (a, b) γp

and (c, d) γ n reactions and for the (a, c) positive and(b, d) negative parities of +.

In summarizing this subsection, we conclude that (i) thedecay distribution for unpolarized + photoproduction is notsensitive to the + parity and (ii) the vector meson dominanceof the background leads to a specific decay distribution whichcan be checked experimentally.

B. Spin observables

1. Single spin observables

Let us consider the beam asymmetry defined as

B = σ (⊥) − σ (‖)

σ (⊥) + σ (‖), (42)

where σ (⊥) and σ (‖) are the cross sections for + photopro-duction with the photon beam polarized perpendicular (εγ =y ) or parallel (εγ = x ) to the production plane, respectively.

We analyze the beam asymmetry as a function of the +polar decay angle . For the resonant channel contribution,σ (⊥) and σ (‖) do not depend on the decay angle if thenucleon spin states are not fixed. Therefore, in this case,the beam asymmetry is a constant and its value depends onthe details of the production mechanism. The interferencewith the background amplitude results in some structure ofthe beam asymmetry. The question is whether or not the beamasymmetries for positive and negative π are different fromeach other at a qualitative level.

In this and subsequent sections, we show the results of ourcalculation for γp → pK0K0 and γ n → nK+K− reactionssystematically for different parity π, different couplingschemes, different signs of α, and different κ∗.

Figures 13 and 14 and Figs. 15 and 16 show the resultsof our calculation for γp → pK0K0 and γ n → nK+K−,respectively. The results for positive and negative π are shown

-1.0 -0.5 0.0 0.5 1.0

cosΘ

0.0

0.5

1.0

W(c

osΘ

)

1/2(+)

tot

BG

Θ+

1/2(-)

(a)(γn)

-1.0 -0.5 0.0 0.5 1.0

cosΘ

0.0

0.5

1.0

W(c

osΘ

)

1/2(+)

tot

(b)

Θ+

1/2(-)

(γp)

BG

-1.0 -0.5 0.0 0.5 1.0

cosΘ

10-4

10-3

10-2

10-1

dσ/d

ΩdM

Θ (

µb/s

r G

eV)

totBG

a2

ρφ

ω

σ

(γp)

f2

(c)

FIG. 12. The decay distributions of (a) nK+ and (b) pK0 in the reactions γ → nK+K− and γ → pK0K0, respectively. The solid anddashed curves correspond to the positive and negative + parities, respectively. The distribution from the background is shown by thedot-dashed curve, whereas the contribution from the pure resonance channel is shown by the solid thin line. (c) The differential cross sectionof the background channels.

035203-11


-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB 1/2

(+)(-)

BG

PV

(a)

+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB 1/2

(+)(-)

BG

PS

(b)

+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2

(+)(-)

BG

PV

(c)

-

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2

(-)

(+)

BG

PS

(d)

-

FIG. 13. The beam asymmetry in γp → pK0K0 as a function of the K decay angle for κ∗ = 0. The results for π = + and π = −are shown in (a, b) and (c, d), respectively. The results for pseudovector (PV) and pseudoscalar (PS) couplings are displayed in (a, c)and (b, d), respectively. The symbol (±) corresponds to positive and negative α. The asymmetries due to the resonant channel are shownby the long dashed (α > 0) and dashed (α < 0) lines, respectively. The asymmetry from the background is shown by the dot-dashedcurves.

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2 PV

(a)

+

0.5 0

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2 PS

(b)

+

0.50

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2 PV

(c)

-

-0.5 0

0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2 PS

(d)

-

-0.5 0

0.5

FIG. 14. The beam asymmetry in γp → pK0K0 as a function of the K decay angle. The result is for α > 0 and κ∗ = 0, ±0.5. Othernotations are the same as in Fig. 13.

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB 1/2

(+)

(-)

BG

PV

(a)

+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB 1/2

(+)(-)

BG

PS

(b)

+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2

(+)(-)

BG

PV

(c)

-

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0Σ

B1/2

(-) (+)

BG

PS

(d)

-

FIG. 15. The beam asymmetry in γ n → nK+K− as a function of the K decay angle. Notation is the same as in Fig. 13.

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2 PV

(a)

+

0.5 0

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2 PS

(b)

+

0.5 0

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2 PV

(c)-

-0.5 0

0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣB

1/2- PS

(d)

-0.5 0

0.5

FIG. 16. The beam asymmetry γ n → nK+K− for different values of κ∗ as a function of the K decay angle. Notation is the same as inFig. 14.

035203-12


in Figs. 13–16 [(a, b) and (c, d), respectively]. The resultsfor the pseudovector (PV) and pseudoscalar (PS) couplingsare displayed in Figs. 13–16 [(a, c) and (b, d), respectively].The asymmetries shown in Figs. 13 and 15 are calculated withκ∗ = 0; the dependence on κ∗ is shown in Figs. 14 and 16.In Figs. 13 and 15, the asymmetries due to the pure resonantchannel are shown by the long dashed (α > 0) and dashed(α < 0) lines. The asymmetry from the background is shownby the dot-dashed curves.

One can see that the resonant channel contribution givesrise to a positive and constant beam asymmetry. For the γp →pK0K0 reaction, its value depends on π: +

B ≈ 2−B ≈

0.6 − 0.8. For the γ n → nK+K− reaction, this dependence israther weak. The asymmetry due to the background channelsis negative with relatively large absolute value. The interplayof the resonant and background processes results in a strongdeviation of B from the constant (“resonant”) values; inall the cases, the asymmetry decreases to negative valueswhen → π . The variation of κ∗ modifies the asymmetryat cos ≈ 1 leaving, however, its shape almost unchanged.In Refs. [8,9], the single beam asymmetry for the reactionγ N → +K was suggested for the determination of π

by measuring the angular distribution of the K mesons. Inour analysis of the two-step process γ N → +K → NKK,however, we find that the dependence of B on the parity π

is not pronounced enough to allow the extraction of the parityof the + state.

To summarize this subsection, we can conclude that thebeam asymmetry cannot be used as a tool for determining π.The shapes of B for positive and negative π are almostthe same. The difference in B in γp → pK0K0 mentionedabove gets small if we compare the result for positive π withκ∗ = 0.5 and the result for negative π with κ∗ = −0.5. Asimilar conclusion is valid for other single spin observables.

2. Double spin observables

As we have seen, the + decay amplitudes in Eq. (5)are related directly to π. Therefore, observables sensitiveto the + parity must involve the spin dependence of theoutgoing nucleon. Let us consider one of them, the target-recoilspin asymmetry, where the spin variables are related to thespin projections of incoming (target) and outgoing (recoil)nucleons,

Atr = σ (↑↑) − σ (↑↓)

σ (↑↑) + σ (↑↓). (43)

Here, σ (↑↑) and σ (↑↓) are the cross sections without and withthe spin flip transition from the incoming to outgoing nucleon,respectively. Using Eqs. (5), one can get the asymmetries dueto the pure resonance channel,

A+tr () = A+

0 cos 2, A−tr () = A−

0 , (44a)

A±0 = dσ±

R (↑↑) − dσ±R (↑↓)

dσ±R (↑↑) + dσ±

R (↑↓), (44b)

where σ+R and σ−

R denote the cross section for + photopro-duction with the positive and negative π values, respectively.

One can see that the spin asymmetries for positive andnegative parities are qualitatively different from one another.In the case of a negative parity, the asymmetry is a constant,independent of . For positive parity, the correspondingasymmetry exhibits a cos 2 dependence which leads to aminimum or a maximum value at = π/2 and a null value at = π/4 and 3π/4.

The next observation is related to the production mecha-nism. The dominance of the K∗ exchange channels leads to arelative suppression of the spin flip transitions for positive π.Therefore, we have A+

0 > 0, which results in a minimum ofthe asymmetry A+

tr = Atrmin at = π/2. For negative π,the dominance of the K∗ exchange channels results in anenhancement of these transitions, so that A−

0 < 0.But this ideal picture is modified when we include the back-

ground contribution. The background processes are dominatedby the spin-conserving transition which results in a positiveasymmetry. Therefore, in the case of a negative π, the totalasymmetry may be either positive or negative. In the case of apositive π, the total asymmetry loses its cos 2 dependence.

Our results for double target-recoil asymmetry are pre-sented in Figs. 17 and 18 for the γp → pK0K0 reactionand in Figs. 19 and 20 for the γ n → nK+K− reaction. Allthe notations are the same as in Figs. 13–16 for the beamasymmetry. For the pure resonance channel contribution, onecan see the cos 2 dependence of Atr for the positive π,and a constant for the negative π. The “modulation” A+

0 forpositive π depends on the tensor coupling κ∗ and decreaseswhen κ∗ increases. The background contribution modifies theasymmetries. In the case of a negative π, the asymmetriesincrease when → π . In the case of a positive π, one cansee a strong modification of the cos 2 dependence. Thismodification is especially strong for larger values of κ∗. Inthis case, we see almost a monotonic decrease of A+

tr as varies from π to 0 [see Figs. 18(a) and 20(a)], similarto the case of the negative π for large and negative κ∗[see Figs. 18(d) and 20(d)]. Therefore, we can conclude thatthe background contribution hampers the use of the doublespin observables for the determination of π because of astrong interplay of the production mechanism (in our exampleit is κ∗) and effects of the + parity in the transitionamplitudes.

3. Triple spin observables

Let us consider the beam asymmetry for the linearlypolarized photon beam at a fixed polarization of the targetand the recoil nucleons. The nucleon polarizations are chosenalong the normal to the production plane [19,21],

yy(↑↑) = σ⊥(↑↑) − σ ‖(↑↑)

σ⊥(↑↑) + σ ‖(↑↑),

(45)

yy(↑↓) = σ⊥(↑↓) − σ ‖(↑↓)

σ⊥(↑↓) + σ ‖(↑↓),

where σ (↑↑) and σ (↑↓) correspond to the spin-conservingand spin flip transitions between the initial and final nucleons,respectively. We choose these asymmetries because for the2 → 2 (γ N → +K) reaction, Bohr’s theorem [46] based on

035203-13


-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(+)

(-)

BG

PV (a)+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(+)

(-)

BG

PS (b)+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(+)(-)

BG

PV

(c)-

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(-)(+)

BG

PS

(d)-

FIG. 17. The double target-recoil spin asymmetry Atr in γp → pK0K0 as a function of the K decay angle. Notation is the same as inFig. 13.

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PV (a)+

0.5

0

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PS (b)+

0.5

0

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PV (c)-

-0.5

0 0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PS (d)-

-0.5

0 0.5

FIG. 18. The double target-recoil spin asymmetry Atr in γp → pK0K0 as a function of the K decay angle for different values of κ∗.Notation is the same as in Fig. 14.

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(+)

(-)

BG

PV (a)+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(+)

(-)BG

PS (b)+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(+)(-)

BG

PV

(c)-

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2

(-)(+)

BG

PS

(d)-

FIG. 19. Same as in Fig. 17, for γ n → nK+K−.

-1.0 -0.5 0.0 0.5 1.0cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PV (a)+

0.50

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PS (b)+

0.5

0

-0.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PV (c)-

-0.5

00.5

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

Atr

1/2 PS (d)-

-0.5

00.5

FIG. 20. Same as Fig. 18, for γ n → nK+K−.

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(+)

(-)

BG

PV

(a)

+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(+)

(-)

BG

PS

(b)

+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(+)

(-)

BG

PV(c) -

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(-)

(+)

BG

PS(d) -

FIG. 21. The triple spin asymmetry yy(↑↑) in γp → pK0K0 as a function of the K decay angle. Notation is the same as in Fig. 13.

035203-14


-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+) 1/2

PV

(a)

+-0.5 0

0.5

1/2-

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+) 1/2

PS

(b)

+-0.5

0

0.5

1/2-

FIG. 22. The triple spin asymmetry yy(↑↑) in γp → pK0K0 asa function of the K decay angle for different values of κ∗ and negativeα. Notation is the same as in Fig. 14.

reflection symmetry in the scattering plane results in

γ N→+Kyy (↑↑) = +π, γ N→+K

yy (↑↓) = −π. (46)

This prediction is very strict; it does not depend on theproduction mechanism (in our case PV- or PS-couplingschemes, α, κ∗, etc.), and therefore it is extremely attractive.But unfortunately, the realistic case is more complicated. Aswe discussed above, the realistic process is the 2 → 3 reaction(γ N → NKK), and we have to take into account the three-bodyaspects of the final state. Let us consider the coplanar reactionwhen all three outgoing particles are in the production planeperpendicular to the nucleon polarization. In this case, Bohr’stheorem predicts

yy(↑↑) = πK = −1, yy(↑↓) = −πK = +1, (47)

independently of the intermediate + parity. It is conceivablethat other “model-independent” predictions made for the 2 →2 reaction may suffer from a similar problem. The only way touse yy as a tool to determine the parity of the + pentaquark isto find a kinematical region where this asymmetry is sensitiveto π and insensitive to the production mechanism.

Consider first the spin-conserving transitions. Figures 21and 22 and Figs. 23 and 24 show results of our calculationof the triple spin asymmetry yy(++) ≡ yy(↑↑) for γp →pK0K0 and γ n → nK+K−, respectively, for different π,different coupling schemes, different signs of α, and differentκ∗. The results for the positive and negative π are shown inFigs. 21 and 23 [(a, b) and (c, d), respectively]. In Figs. 22 and24, the results for both parities are displayed simultaneously.The results for pseudovector (PV) coupling are shown in panels(a) and (c) of Figs. 21 and 23 and panels (a) of Figs. 22 and 24;those for pseudoscalar (PS) coupling are given in panels (b)

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+) 1/2

PV (a)

+

-0.5

00.5

1/2-

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+) 1/2

PS (b)

+

-0.5

0

0.5

1/2-


and (d) of Figs. 21 and 23 and panels (b) of Figs. 22 and 24.The asymmetries shown in Figs. 21 and 23 are calculated withκ∗ = 0; the dependence on κ∗ is shown in Figs. 22 and 24. InFigs. 21 and 23, the asymmetries due to the resonant channelare shown by the solid curves (α > 0) and the dashed (α < 0)curves. The asymmetry due to the background is shown by thedot-dashed curves.

The case of = 0, π corresponds to the coplanar geometry[Eq. (47)], where yy(↑↑) = −1 independently of π, thereaction mechanism, and input parameters. For the negative+ parity, the asymmetry due to the resonant channel remains−1 at all cos because of the s-wave + decay, and thereforepredictions for 2 → 2 and 2 → 3 for this case are the same.

For positive parity, we have a p-wave decay which leadsto a fast increasing asymmetry from −1 up to positive andlarge values and results in a specific bell-shape behaviorof yy . In principle, the shapes of yy for different π

are quite different from each other in the region of −0.8 <∼cos <∼ 0.8 and practically do not depend on the productionmechanism. Therefore, one could consider using this asym-metry for determining π. Unfortunately, as in the case ofthe double target-recoil asymmetry, this picture is modifiedby the interference between the resonance and backgroundchannels. For negative π and negative α, one can see asizable deviation of yy(↑↑) from −1 at negative cos . Theeffect of the resonance-background interference is large in theγ n → nK+K− reaction, leading to a decreasing +

yy(↑↑) atcos ≈ 0 and an increasing −

yy(↑↑) at cos ≈ 0–0.2 ascompared to that in the γp → pK0K0 reaction. Nevertheless,from Figs. 22 and 24, one can see that in the region of0.5 <∼ cos <∼ 0.8, the asymmetries ±

yy(↑↑) for differentparities are qualitatively different from each other. This feature

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(+)(- )

BG

PV(a)+

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(+)(- )

BG

PS(b) +

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(+)

(- ) BG

PV(c) -

-1.0 -0.5 0.0 0.5 1.0

cosΘ

-1.0

-0.5

0.0

0.5

1.0

ΣY

Y(+

+)

1/2

(- )(+)

BG

PS(d) -


035203-15


-1.0 -0.5 0.0 0.5 1.0

cosΘ

0.4

0.6

0.8

1.0

ΣY

Y(+

-)

1/2

PV

(a)

+

-0.5

0

0.5

1/2-

(γp)

-1.0 -0.5 0.0 0.5 1.0

cosΘ

0.4

0.6

0.8

1.0

ΣY

Y(+

-)

1/2

PV

(b)

+

-0.5

0

0.5

1/2-

(γn)

FIG. 25. The triple spin asymmetry yy(↑↓) in (a) γp →pK0K0 and (b) γ n → nK+K− as a function of the K+ decay anglefor different values of κ∗ and positive α and the PV-coupling scheme.

suggests the use of this observable for the determinationof π.

Consider now the spin flip transitions. For the vectormeson photoproduction, the spin flip transitions are suppressedcompared to the spin-conserving ones, and therefore, the effectof the background in ±

yy(↑↓) is much smaller than in ±(↑↑).Now, −

yy(↑↓) is close to its “model-independent” limit (+1)[cf. Eq. (47)]. For the positive parity, we expect +

yy(↑↓) = +1at cos = ±1 with some decrease at small |cos |. Theresults of the calculations of yy(+−) ≡ yy(↑↓) shownin Figs. 25(a, b) for γp → pK0K0 and γ n → nK+K−,respectively, confirm this expectation. They are obtainedusing the PV-coupling scheme and different κ∗. The resultscorresponding to other input parameter values are similar tothose shown in Figs. 25(a) and (b), and we do not display themhere. One can see a clear difference between the predictions forpositive and negative π. For negative parity, −

yy(↑↓) ≈ +1at all cos . For positive parity, we have +

yy(↑↓) = +1 atcos = ±1 with a minimum at cos = 0. Unfortunately, theminimum value of +

yy(↑↓) depends strongly on the tensorcoupling κ∗; in particular, at large negative κ∗, the deviationbetween +

yy(↑↓) and −yy(↑↓) may be rather small. This

makes it difficult to use yy(↑↓) for the determination of π.

VI. SUMMARY

We have analyzed in detail for the first time two essentialaspects of the problem associated with the determination of theparity of the + pentaquark from photoproduction processes.The first one is the nonresonant background in the reactionγ N → NKK. The second one is related to the three-body finalstates. The interference between the resonance amplitude andthe nonresonant background results in a modification of allthe spin observables from those corresponding to the pureresonance channels contribution. The effect of the three-bodyfinal state means that the predictions made for the spin “observ-ables” of the (directly unobservable) two-body intermediatereaction γ N → +K are not useful from a practical point ofview. This applies to all “model-independent” predictions forthe + parity determination in + photoproduction discussedrecently in the literature [19–21].

We have analyzed in detail the nonresonant background.We have found that in the near-threshold energy region(Eγ ≈ 2 GeV), the nonresonant background is dominatedby the vector (φ, ρ, and ω) meson photoproduction. Thecontributions from the scalar (σ ) and tensor (a2, f2) mesonsare rather small. However, the latter may be important athigher energies. Additional information about the backgroundstructure may be found by studying the relative angulardistribution of the KK pair. In our study, we have shown that,using both the measured ratio of the resonance to backgroundyields and the calculated nonresonant background, we canfind the strength of the K∗ exchange amplitude and reduce thenumber of unknown parameters.

The present paper aims, as mentioned in the introduction,at showing whether it is possible to determine the +parity in the reaction γ N → NKK by taking into account thefinite + decay width and interference between resonanceand nonresonance channels and ambiguities of productionmechanism. Among the observables investigated in this work,the triple spin asymmetry shows a kinematical “window”where the predictions are sensitive to the + parity and ratherinsensitive to the various reaction mechanisms considered,and thus, this observable offers a possibility of determiningthe parity of the +.

The present analysis is based on the observed ratios ofthe resonant and nonresonant contributions, including theinstrumental energy resolution. Improvement of the instru-mental resolution in the near future will increase the ratio, andaccordingly the difference between the triple spin asymmetries(↑↑) for the positive and negative π will get quiteconspicuous, and thus will provide the + parity almost modelindependently.

Experimental measurements of the triple spin asymmetrywill require polarized photons, polarized target nucleons, andpolarization measurement of the residual nucleon spin. TheLEPS facility provides fully polarized photons, and the HDtarget provides the polarized nucleons. Measurements of theresidual nucleon polarization may be carried out by doublescattering experiments with intense photon beams, which willbe realistic with the top-up operation of the electron ring. Weleave to future experimental papers detailed discussions onpractical methods, which are beyond the scope of the presenttheoretical study.

ACKNOWLEDGMENTS

We thank S. Date, K. Hicks, A. Hosaka, M. Fujiwara,T. Mibe, T. Nakano, Y. Oh, Y. Ohashi, and H. Toki for fruitfuldiscussion. One of the authors (A.I.T.) thanks T. Tajima,the director of the Advanced Photon Research Center, JapanAtomic Energy Research Institute, for his hospitality duringAIT’s stay at SPring-8.

APPENDIX A: TRANSITION OPERATORS FORRESONANCE CHANNELS

We show here the explicit expressions for the transitionoperators Mµ in Eqs. (11) for the positive and negative +parity (π) and for the PV- and PS-coupling schemes.

035203-16


The specific parameters for the form factor of Eq. (10)required here are defined by

Fs = F (MN, s), Fu = F (M, u), and

Ft = F (MK+ , t). (A1)

In addition, we need the form-factor combinations

Ftu = Ft + Fu − FtFu and Fsu = Fs + Fu − FsFu

(A2)

to construct the contact terms Mcµ given below that make the

initial photoproduction amplitude gauge invariant [28,29]. Thefour-momenta in the following equations are defined accordingto the arguments given in the reaction equation

γ (k) + N (p) → +(p) + K(q). (A3)

A. γ n → + K−

π = +1; PV

Mtµ = i

egNK (k − 2q)µγ5

t − M2K+

Ft , (A4a)

Msµ = i

egNK

M + MN

γ5q/p/ + k/ + MN

s − M2N

(i

κn

2MN

σµνkν

)Fs,

(A4b)

Muµ = i

egNK

M + MN

(γµ + i

κ

2M

σµνkν

)× p/ − k/ + M

u − M2

γ5q/ Fu, (A4c)

Mcµ = iegNKγ5

[(k − 2q)µt − M2

K+(Ftu − Ft ) + (2p − k)µ

u − M2

× (Ftu − Fu) + γµ

M + MN

Fu

]. (A4d)

π = +1; PS

Mtµ = i

egNK (kµ − 2qµ)γ5

t − M2K+

Ft , (A5a)

Msµ = iegNKγ5

p/ + k/ + MN

s − M2N

(i

κp

2MN

σµνkν

)Fs,

(A5b)

Muµ = iegNK

(γµ + i

κ

2M

σµνkν

)p/ − k/ + M

u − M2

γ5 Fu,

(A5c)

Mcµ = iegNKγ5

[(k − 2q)µt − M2

K+(Ftu − Ft ) + (2p − k)µ

u − M2

×(Ftu − Fu)

]. (A5d)

For both PS and PV couplings, the positive-parity t-channelK∗ exchange is given by

Mtµ(K∗) = egγKK∗gNK∗

MK∗

εµναβkαqβ

t − M2K∗

×[γ ν − i

σ νλ(p − p)λM + MN

κ∗]

F (MK∗, t). (A6)

For the negative + parity, we have the following ampli-tudes.

π = −1; PV

Mtµ = i

egNK (k − 2q)µt − M2

K+Ft , (A7a)

Msµ = −i

egNK

M − MN

q/p/ + k/ + MN

s − M2N

(i

κn

2MN

σµνkν

)Fs,

(A7b)

Muµ = −i

egNK

M − MN

(γµ + i

κ

2M

σµνkν

)×p/ − k/ + M

u − M2

q/ Fu, (A7c)

Mcµ = iegNK

[(k − 2q)µt − M2

K+(Ftu − Ft ) + (2p − k)µ

u − M2

× (Ftu − Fu) − γµ

M − MN

Fu

]. (A7d)

π = −1; PS

Mtµ = i

egNK (k − 2q)µt − M2

K+Ft , (A8a)

Msµ = iegNK

p/ + k/ + MN

s − M2N

(i

κn

2MN

σµνkν

)Fs, (A8b)

Muµ = iegNK

(γµ + i

κ

2M

σµνkν

)p/ − k/ + M

u − M2

Fu,

(A8c)

Mcµ = iegNK

[(k − 2q)µt − M2

K+(Ftu − Ft ) + (2p − k)µ

u − M2

×(Ftu − Fu)

]. (A8d)

For negative + parity, the K∗ exchange is given by

Mtµ(K∗) = egγKK∗gNK∗

MK∗

εµναβkαqβ

t − M2K∗

γ5

×[γ ν − i

σ νλ(p − p)λM + MN

κ∗]

F (MK∗, t).

(A9)

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B. γ p → + K 0

π = +1; PV

Msµ = i

egNK

M + MN

γ5q/p/ + k/ + MN

s − M2N

×(

γµ + iκp

2MN

σµνkν

)Fs, (A10a)

Muµ = i

egNK

M + MN

(γµ + i

κ

2M

σµνkν

)× p/ − k/ + M

u − M2

γ5q/ Fu, (A10b)

Mcµ = i

egNK

M + MN

γ5q/

[(2p + k)µs − MN

(Fsu − Fs)

+ (2p − k)µu − M2

(Fsu − Fu)

]. (A10c)

π = +1; PS

Msµ = iegNKγ5

p/ + k/ + MN

s − M2N

(γµ + i

κp

2MN

σµνkν

)Fs,

(A11a)

Muµ = iegNK

(γµ + i

κ

2M

σµνkν

)× p/ − k/ + M

u − M2

γ5 Fu, (A11b)

Mcµ = iegNKγ5

[(2p + k)µs − MN

(Fsu − Fs)

+ (2p − k)µu − M2

(Fsu − Fu)

]. (A11c)

The transition amplitudes for the negative + parity may beobtained immediately from Eqs. (A10a)–(A10c) and (A11a)–(A11c) by the simple substitutions

PV:γ5

M + MN

→ − 1

M − MN

; (A12a)

PS: γ5 → 1. (A12b)

APPENDIX B: POMERON EXCHANGE AMPLITUDE

The invariant amplitude for the Pomeron exchange processhas the form

APf i = −MP (s, t) P

f i. (B1)

The scalar function MP (s, t) is described by the Reggeparametrization,

MP (s, t) = CP F1(t) F2(t)1

s

(s

sP

)αP (t)

exp

[− iπ

2αP (t)

],

(B2)

TABLE II. Coupling constants for the nonresonant background.

g Value Source Ref.

gπNN 13.26 φ,ω photoproduction [32,34]gηNN 3.54 SU(3) [30]gσNN 10.03 ρ photoproduction [32]gρNN ; κρ 3.72; 6.71 Bonn model [38]egρσγ 0.82 ρ photoproduction [32]egρπγ 0.16 ρ photoproduction [32]egωπγ 0.55 ω photoproduction [32]egφπγ −0.043 φ → πγ decay, SU(3) [41]egφηγ −0.214 φ → ηγ decay, SU(3) [41]g0 ≡ gφKK 4.48 φ → KK decay [41]gρK+K− g0/

√2 SU(3) [41]

gρK0K0 −g0/√

2 SU(3) [41]gσKK = −gσππ 1.74 σ → ππ decay, SU(3) [41]

where F1(t) is the isoscalar electromagnetic form factor of thenucleon, and F2(t) is the form factor given in Appendix C forthe vector-meson–photon–Pomeron coupling. We also followRef. [48] to write

F1(t) = 4M2N − 2.8t(

4M2N − t

)(1 − t/t0)2

,

(B3)

F2(t) = 2µ20(

1 − t/M2V

)(2µ2

0 + M2V − t

) ,

where t0 = 0.7 GeV2. The Pomeron trajectory is known to beαP (t) = 1.08 + 0.25 t . The strength factor CP is given by

CP = 6eg2

γV

, (B4)

where γV is the vector meson decay constant (γ 2ω =

72.71, γ 2φ = 44.22, and γ 2

ρ = 6.33). The parameter g has ameaning of the Pomeron-quark coupling and it is taken to bethe same for all vector mesons (g2 = 16.6). The remainingparameters read µ2

0 = 1.1 GeV2 and sP = 4 GeV2. Theamplitude P

f i reads

Pf i = uf k/ ui

(ε∗λV

· ελγ

) − uf ε/λγui

(ε∗λV

· k) − uf ε/∗

λVui

×[ελγ

· q −(ελγ

· p)(k · q)

p · k

], (B5)

with p = (p + p′)/2. εµ(V ) and εν(γ ) are the polarizationvectors of the vector meson (ρ, φ) and the photon, respectively,and ui = umi

(p) [uf = umf(p′)] is the Dirac spinor of the

nucleon with momentum p (p′) and spin projection mi (mf ).

APPENDIX C: PARAMETERS FOR THENONRESONANT BACKGROUND

The parameters of the model that define the amplitude ofthe vector meson photoproduction are taken from previousstudies [32,34]. The coupling constants in Eqs. (12a)–(12e)are based on empirical knowledge, comparison with the cor-responding decay widths and SU(3) symmetry considerations.For the σ -meson photoproduction, we use the Bonn model

035203-18


as listed in Table B.1 (Model II) of Ref. [38]. All couplingconstants are displayed in Table II.

All the vertex functions are dressed by monopole formfactors. We use as expressions for their products

Fπ (t) =(

0.62 − m2π

0.62 − t

)2

, (C1a)

Fη(t) =(

1.02 − m2η

1.02 − t

) (0.92 − m2

η

0.92 − t

), (C1b)

Fσ (t) =(

1.02 − m2σ

1.02 − t

) (0.92 − m2

σ

0.92 − t

), (C1c)

Fρ(t) =(

1.32 − m2ρ

1.32 − t

)2

, (C1d)

where all the masses are in GeV and t is in GeV2.

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parity determination in the reaction

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