c Pleiades Publishing, Ltd., 2008. ISSN 1063-7788, Physics of Atomic Nuclei, 2008, Vol. 71, No. 8, pp. 1410–1423.
ELEMENTARY PARTICLES AND FIELDS Theory
On the K +d Interaction at Low Energies* V. E. Tarasov1), V. V. Khabarov1), A. E. Kudryavtsev1), and V. M. Weinberg1), 2) Received August 9, 2007
Abstract—The Kd reactions are considered in the impulse approximation with N N final-state interactions (FSI) taken into account. Realistic parameters for the KN phase shifts are used. The “quasi-elastic” energy region, in which the elementary KN interaction is predominantly elastic, is considered. The theoretical predictions are compared with the data on the K + d → K + pn, K + d → K 0 pp, K + d → K + d, and K + d total cross sections. The N N FSI effect in the reaction K + d → K + pn has been found to be large. The predictions for the Kd cross sections are also given for slow kaons, produced from φ(1020) decays, as the functions of the isoscalar KN scattering length a0 . These predictions can be used to extract the value of a0 from the data. PACS numbers: 13.75.Jz, 24.10.-i, 25.10.+s, 25.80.Nv DOI: 10.1134/S1063778808080127
1. INTRODUCTION Recent experimental indications [1] of the possible existence of the exotic Θ+ (1540) state in the K + N system (pentaquark s¯uudd state) have enhanced the interest in the K + N and K + d interactions. Analyses of the K + N elastic scattering [2, 3], K + d total cross sections [4, 5], and the reaction K + d → K 0 pp [6, 7] with isospin I = 0 assumed for the pentaquark led to a conclusion about its small width Γ ≤ 1 MeV. At the present time, the existence of the pentaquark seems to be doubtful since it was not confirmed by a number of experiments (see the review paper by Danilov and Mizuk listed in [1]). In connection with the Θ+ problem, the recent paper [8] introduces a new partial-wave analysis (PWA) of K + N scattering in the momentum range 0 < plab < 1.5 GeV/c and comparison of the results with previous analyses. To extract the K + N -scattering parameters in the isospin-0 channel, additional information to the data on the proton target is required. This additional data are provided by experiments on the K + d collisions. The recent paper [9] contains a review of existing data on the K + d reactions in the “quasi-elastic” region plab < 0.8 GeV/c (where the elementary KN reaction is predominantly elastic and the role of the particle-production processes is negligible), i.e., on the processes K + d → KN N ∗
The text was submitted by the authors in English. Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia. 2) Moscow Institute of Physics and Technology, Dolgoprudny, Russia. 1)
and K + d → K + d. A reasonable description of the total cross sections and differential spectra dσ/dΩ for outgoing kaons was obtained in the framework of ¨ the impulse approximation with the use of the Julich model for the KN amplitude. The papers [8, 9] also claim that the existing K + N and K + d data do not prove, but do not exclude, the possibility of a narrow resonance Θ+ in the KN (I = 0) system. In this paper, we present calculations of the total K + d cross sections and the cross sections of the breakup reactions K + d → K + pn, K 0 pp and of the elastic scattering process K + d → K + d. We restrict our consideration to the quasi-elastic region plab < 0.8 GeV/c defined above. We consider only the “background” amplitudes and neglect the possible Θ+ contribution. In our calculations, we take into account the pole diagrams and s-wave final-state interaction (FSI) of slow nucleons in the process Kd → KN N . We express the KN -scattering (and chargeexchange) amplitude through the partial-wave components, including only s- and p-wave terms which are important in the region of interest. For simplicity, we use the s-wave KN -scattering amplitude when calculating the N N FSI contribution. In this approximation, the KN amplitude does not contain the spin-flip term and as a result the N N FSI is taken into account only in the triplet N N (3 S1 ) state. Thus, according to the Pauli principle, in the case under consideration N N FSI takes place only for the process K + d → K + pn, but not for K + d → K 0 pp. Predictions will also be given for the K + d cross sections with slow kaons as functions of the isoscalar KN -scattering length a0 . They can be used to extract
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s-Wave scattering lengths aI ≡ a0 and p-wave scattering volumes vI± ≡ a1± from [5, 8] (I)
(I)
Ref.
a1 , fm
v1− , fm3
v1+ , fm3
[5]
–0.328
–0.02
0.015
[8]
–0.308
–0.092
0.103
the value of a0 from experiments with kaons produced from φ(1020) decays at rest. The corresponding experiments with φ(1020) mesons produced in e+ e− collisions may be proposed for the DAΦNE machine (Frascatti). The paper is organized as follows. In Section 2, we give the expressions for the partial-wave KN -scattering amplitudes and illustrate the description of the data on the KN cross sections for some sets of phase-shift parameters. In Section 3, we write out the amplitudes for the breakup reactions Kd → KN N (pole diagrams + N N FSI) and the elastic process K + d → K + d. In Section 4, our theoretical predictions for the K + d cross sections are presented and compared with the experimental data. Section 5 is the conclusion. Some necessary formulas are placed in the Appendix. 2. KN -SCATTERING AMPLITUDE Let us write out the isospin structure of the KN -scattering amplitude fˆKN and the relations between charge and isospin amplitudes. They read 1 fˆKN = FˆS + FˆV τK τ , FˆS = (3Fˆ1 + Fˆ0 ), (1) 4 1 FˆV = (Fˆ1 − Fˆ0 ), fˆK + p = Fˆ1 , 4 1 1 fˆK +n = (Fˆ1 + Fˆ0 ), fˆK +n→K 0 p = (Fˆ1 − Fˆ0 ), 2 2 where τ (τK ) are the isospin Pauli matrices for nucleons (kaons) and Fˆ0,1 (FˆS,V ) are the KN amplitudes with s-channel (t-channel) isospin I = 0, 1. The K + N -scattering and charge-exchange amplitudes are3) fˆK + N →K +N = FˆS + FˆV τ3 , (2) fˆK +n→K 0p = 2FˆV τ+ , 3)
1 τ+ = (τ1 + iτ2 ). 2
With these amplitudes fˆ, the differential cross section for ˆ 2 binary reaction is dσ/dΩ = |ϕ+ 2 f ϕ1 | q2 /q1 . Here, ϕ1,2 are the spinors (and isospinors) of the initial and final nucleon and ϕ+ i ϕi ≡ 1; q1,2 are the initial and final relative momenta. Throughout the paper, we use the word “amplitude” for the ˆ ˆ matrix element ϕ+ 2 f ϕ1 and for the operator f as well. PHYSICS OF ATOMIC NUCLEI Vol. 71
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v0− , fm3
v0+ , fm3
–0.06
0.123
–0.010
–0.1048
0.183
–0.029
a0 , fm
The amplitudes FˆI (I = 0, 1) can be written in standard form k × k , (3) FˆI = AI + BI (nσ), n = |k × k | ∞ (I) (I) (l + 1)fl+ + lfl− Pl (cos θ), AI = l=0
∞ (I) (I) fl+ − fl− Pl1 (cos θ), BI = i l=1
η exp(2iδ) − 1 . 2ik Here, k (k ) is the relative initial (final) c.m. mo(I) I are the inelasticities mentum; η ≡ ηl± and δ ≡ δl± (0 ≤ η ≤ 1) and phase shifts for s-channel isospin I, orbital momentum l, and j = l ± 1/2. At plab < 0.8 GeV/c the cross sections of particle production in (I) the KN interactions are relatively small and ηl± ≈ 1. (I)
fl± =
(I)
(I)
Hereafter, we take ηl± ≡ 1 and fl± = 1/(k cot(δ) − ik). Let us compare the KN cross sections calculated I with the data. Here, from the existing phase shifts δl± we use the phase shifts from [5, 8] given in the forms 1 I ) = (I) (l = 0, 1) [5], (4) k cot(δl± al± k2l I )= k cot(δl±
1 (I)
al± k2l
(I) bi,l± k2i 1+
(5)
i≥1
(l = 0, 1) [8], (I)
(I)
(I)
(I)
δ2± = k5 (c± + d± k2 + e± k4 ), (I)
(I)
δ3± = k7 f± (I)
(I)
(I)
(I)
[8],
(I)
(I)
where al± , bi,l± , c± , d± , e± , and f± are constants. Their values were obtained [8] from PWA of K + N scattering in the range plab < 1.5 GeV/c, using s-, p-, d-, and f -wave amplitudes (see Tables II and (I) VI in [8]). The values al± for l = 0, 1 (scattering lengths and volumes) from [5, 8] are given in the table. Figure 1 shows the total K + p, K + n, and + K N (I = 0) cross sections. The symbols represent
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σ, mb 20 K+p
σ, mb 25 (a)
(c)
K+N (I = 0)
15 20
10 [10, 11] [15]
[12] [16]
[13] [17]
15
[14] [18]
5 20 K+n
(b)
10
15
10 5 [10, 11] [13] [21]
5 [19] [20] 0
0.2
0.4
0.6
0.8 plab, GeV
0 0.2
0.4
0.6
0.8 plab, GeV
Fig. 1. The total (a) K + p, (b) K + n, and (c) K + N (I = 0) cross sections. The curves correspond to different sets of KN phase shifts (see text), and the symbols correspond to the experimental data. The closed (open) symbols in Fig. 1a show the data on the total (elastic) K + p cross sections.
(a)
(b) k1
k
k1
k p3
p1 ( p2)
p1 p4
p2 ( p1)
P M1 (M2)
p2
P MR
Fig. 2. (a) The pole (M1 and M2 ) and (b) the N N FSI (MR ) diagrams for the Kd → KN N process. The solid, dashed, and double lines correspond to the kaons, nucleons, and deutrons, respectively.
the experimental data taken from [10–21]. Figure 1a t el shows the data on the total (σK + p ) and elastic (σK + p )
K + p cross sections. The curves in Figs. 1a and 1b t el show the calculated values σK + p = σK + p (a) and
t el σK + n = σK + n + σK + n→K 0 p (b). The cross sections I=0 σKN in Fig. 1c were calculated through the relation I=0 = 2σ t t σKN K + n − σK + p in accordance with the data from [10, 11, 13]. The solid and dotted curves in Fig. 1 correspond
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to the results obtained with the use of parameters from [8] and show the contributions of the partial waves with 0 ≤ l ≤ 3 and 0 ≤ l ≤ 1, respectively. As one can see from Fig. 1, the d- and f -wave contributions are negligible at plab < 0.7 GeV/c. The dashed curves show the results obtained with the phase shifts from [5], where only s and p waves were included. Below, calculating the Kd cross sections, we use the KN parameters from [8].
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M1 and M2 , we find it convenient to rewrite one of ˆ c them, say M2 , with the help of identity ϕ+ 2 Aϕ1 ≡ + ˆc c ϕ1 A ϕ2 , where Aˆ is an arbitrary operator, containing the Pauli spin (σ) and isospin (τ ) matrices, and Ac ≡ σ2 τ2 AT σ2 τ2 (note that I c = I, σc = −σ, and τ c = −τ ). Then we can rewrite the amplitudes (7) as ˆ 2 )ϕc , (8) M1 = c1 ϕ+ fˆKN Ψ(p 1 2 1 + ˆc c −c2 ϕ1 Ψ (p1 )fˆKN2 ϕc2 ,
M2 = where ci = 16π msKNi . The expressions for the 3. AMPLITUDES OF THE Kd REACTIONS particle momenta used to calculate the amplitudes 3.1. The Pole Amplitudes of the Breakup Reactions are given in Section 2 of Appendix. The squares The pole diagrams for the Kd → KN N reactions and interferences of the amplitudes for the reactions are shown in Fig. 2a, where M1 and M2 also stand Kd → KN N with unpolarized particles are given in 5) for the corresponding invariant amplitudes (the par- Section 3 of Appendix. ticle momenta are given in the deuteron rest frame). Strictly speaking, in the case of the reaction Hereafter, we consider the deuteron and nucleons as K + d → K + pn, the pole amplitudes M1,2 (8) should nonrelativistic particles and kaons as relativistic ones. also contain a term proportional to the Coulomb To calculate the Kd amplitudes, we use the deuteron K + p-scattering amplitude f . It can be included by C wave function (DWF) Ψ(q) in the form 1 the replacement fˆKNi → fˆKNi + fC (1 + τ3 ) in the 1 +ˆ 2 (6) hadronic K + N -scattering amplitudes. Since f ∼ Ψ(q) = √ ϕ2 Ψ(q)ϕc1 , C 2 1/t, where t = (k − k1 )2 is the square of the fourw(q) 3(q)(qσ) u(q) ˆ momentum transfer, the Coulomb effects may be − (σ) Ψ(q) = √ (σ) − 2 q2 2 essential at small scattering angles of the outgoing the measured cross sections (q = |q|). Here, ϕ1,2 are the nucleon spinor–isospi- kaons (small t). Thus, + d → K + pn should depend on the of the reaction K c nors (ϕ+ i ϕi ≡ 1); the notation ϕ means the chargeexperimental conditions. c conjugated spinor–isospinor ϕ ≡ τ2 σ2 ϕ∗ ; q and Note that, in the breakup reaction Kd are the relative momentum of the nucleons and polar√ → KN N , < 0, since sN N = md we always have t ≤ −|t| min ization vector of the deuteron, respectively; u(q) and √ (m and s are the deuteron mass and the efN N d w(q) are the s- and d-wave of the DWF, respec 3 parts fective mass of the N N system). Thus, the total 2 2 3 tively, normalized as d q[u (q) + w (q)] = (2π) . Coulomb cross section σC (K + d → K + pn) is finite, The invariant amplitudes M1 and M2 with DWF (6) unlike the case of the elastic-scattering processes read4) for which it diverges at zero scattering angles (at √ + ˆ c ˆ M1 = 2 mϕ1 MKN1 Ψ(p2 )ϕ2 , (7) t → 0). An estimate of the Coulomb cross section √ σC (K + d → K + pn) is given in Section 6 of Appendix. c ˆ ˆ M2 = −2 mϕ+ 2 MKN2 Ψ(p1 )ϕ1 . In the following, we neglect the Coulomb effects when Here, m is the nucleon mass; ϕi is the spinor–isospi- calculating the Kd cross sections. ˆ KN = 8π √sKN fˆKN nor of the ith final nucleon; M i i i is the KN -scattering (on the ith nucleon) invari3.2. The Final-State Interaction of Nucleons √ ant amplitude; sKNi is the invariant mass of the At low energies, the important effect should come KNi system. The cross sections, expressed through from the N N FSI owing to large N N -scattering the invariant amplitudes, are given in Section 1 of lengths. The famous FSI effect is responsible for Appendix. The amplitude M2 in (7) can be obtained the near-threshold enhancement in the mass specfrom M1 by interchanging the nucleons, i.e., M2 = tra dσ/d√s N N (Migdal–Watson effect [24, 25]) of −M1 (N1 ↔ N2 ). Thus, the amplitude M1 + M2 is the meson production reactions N N → N N x and antisymmetric with respect to transposition of nucleons in accordance with the Pauli principle. Fur5) There are relations, derived in [23], for the differential cross ther, calculating the interference of the amplitudes sections of reactions Kd → KN N in the impulse approxi4)
When writing out the amplitudes of the reactions on the deuteron, we follow the diagrammatic technique of [22]. PHYSICS OF ATOMIC NUCLEI Vol. 71
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mation with spin and isospin variables taken into account. However, they cannot be applied when the d-wave part of DWF or the rescatterings are included.
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increases the reaction cross section in the nearthreshold region. The role of secondary rescatterings was also investigated in reactions on the deuteron (for example, in the reaction π − d → π − pn [26]). In this paper, we take into account only the N N rescattering amplitudes and neglect the KN rescatterings, since the KN -scattering lengths are relatively small, aKN aN N . The N N FSI diagram is shown in Fig. 2b (the diagram MR ). We consider only the s-wave N N rescattering. It is convenient to write the invariant amplitude of the s-wave scattering N3 N4 → N1 N2 in the form √ (S) (9) MN N = 8π sN N
σ σ c (0) ϕ+ × fN N (p) ϕc+ 4 √ ϕ3 1 √ ϕ2 2 2
τ (1) + τ c √ √ ϕ ϕ ϕ , + fN N (p) ϕc+ 3 2 4 1 2 2 √ where sN N is the invariant mass of the N N system √ (we take sN N = 2m for the nonrelativistic nucle(I=0,1)
ons) and fN N (p) are the N N -scattering amplitudes with isospin I and relative momentum p, normalized as dσ/dΩ = |f |2 . The first (second) term in Eq. (9) corresponds to the N N -scattering amplitude with isospin I = 0 (1) and total spin SN N = 1 (0) in accordance with the Pauli principle. Let the N3 N4 → N1 N2 amplitude be given in the + ˆ c ˆ general form MN N = (ϕc+ 4 B2 ϕ3 )(ϕ1 B1 ϕ2 ), where ˆ1,2 contain the nucleon spin (σ) and the operators B isospin (τ ) Pauli matrices. Then, making use of the ˆ KN from Eqs. (6) and (7), one ˆ i ) and M notation Ψ(p can obtain the N N FSI amplitude MR in the form −1 dp4 (10) MR = √ m (2π)3 ˆ2 M ˆ KN } ˆ 4 )B Tr{Ψ(p ˆ c (ϕ+ × 1 B1 ϕ2 ). 2 2mε3 − p3 + i0 Here, pi and εi (i = 1, 2, 3, 4) are the nucleon momenta in the deutron rest frame (see Fig. 2b) and kinetic energies, respectively; the nucleon with the momentum p4 is “on-shell”; ε3 = ε1 + ε2 − ε4 ; ε1,2,4 = ˆ where Tr{Tˆ} p21,2,4 /2m; Tr{(. . .)} ≡ Tr{Tˆ}Tr{S}, ˆ is the trace of isospin (spin) part Tˆ (S) ˆ of (Tr{S}) ˆ Taking the N N the matrix expression (. . .) = TˆS. amplitude, given by Eq. (9), we should make the replacement ˆ KN }(ϕ+ B ˆ c ˆB ˆ2 M (11) Tr{Ψ 1 1 ϕ2 ) (0)off ˆ M ˆ KN }(ϕ+ σϕc2 ) → 8πm fN N (q, p)Tr{Ψσ 1
(1)off ˆ M ˆ KN }(ϕ+ τ ϕc2 ) + fN N (q, p)Tr{Ψτ 1 (I)off
in Eq. (10). Here, fN N (q, p) is the “off-shell” N N amplitude with isospin I and q(p) is the relative momentum of the intermediate (final) nucleons in the diagram MR . To simplify the calculations, we take into account only the s-wave KN scattering in the amplitude MR . We shall comment on this approximation ˆ KN below in Section 4. In this case, the operator M ˆ M ˆ KN } = 0, contains no spin matrices σ and Tr{Ψτ ˆ ∼ Tr{σ} = 0. It means since the spin trace Tr{Ψ} (1) that the term proportional to fN N vanishes in the amplitude MR . Thus, in our approximation, the N N FSI takes place only in the triplet N N (3 S1 ) state with isospin I = 0 in the reaction K + d → K + pn and is absent in the reaction K + d → K 0 pp. For the first term of the right-hand part of Eq. (11), we get ˆ M ˆ KN }(ϕ+ σϕc ) (12) Tr{Ψσ 2 1 √ (1) (0) ˆ c = 8π sKN 3f0 + f0 (ϕ+ 1 Ψϕ2 ) (here an additional factor of 2 comes from isospin (I) trace Tr{I} = 2), where f0 are the s-wave KN amplitudes with isospins I = 0, 1 [see (3)]. (0)off
The off-shell N N amplitude fN N (q, p) is chosen here in the form corresponding to the scattering on the separable Yamaguchi potential. For the on(0) shell amplitude fN N (p), we use the known parameters [27]. Then (0)off
fN N (q, p) = (0)
fN N (p) = β ≈ 240 MeV,
p2 + β 2 (0) f (p), q2 + β 2 N N 1 (0)
1/aN N + 12 r0 p2 − ip (0)
aN N = −5.4 fm,
(13) ,
r0 = 1.7 fm.
Finally, applying Eqs. (10)–(13), we obtain the amplitude MR in the form + ˆ 2 2 ˆ (14) MR = cAR ϕ1 L(−p ) − L(β ) ϕc2 , 1 (0) (1) (0) AR = fN N (p) 3f0 + f0 , 4 ˆ + ∆) dq Ψ(q ˆ , L(x) = 8π (2π)3 (q 2 + x − i0) p1 − p2 p1 + p2 , p= , ∆= 2 2 √ where c = 16π msKN . Here, we evaluate the KN (I)
amplitudes f0 for the target nucleon at rest in the ˆ deuteron rest frame. The integral L(x) is calculated PHYSICS OF ATOMIC NUCLEI Vol. 71 No. 8
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ON THE K + d INTERACTION
in Section 4 of Appendix, where the analytical expression is given in the case of the Bonn [28] or Paris [29, 30] DWF.
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3.3. Elastic (Coherent) K + d Scattering Close to threshold, the contribution of the elastic process K + d → K + d dominates in the total cross t . We shall use a single-scattering approxsection σKd el imation for the elastic-scattering amplitude MK +d (see the diagram in Fig. 3), neglecting relatively small contributions of kaon rescattering and meson exchange currents [31] to the integrated cross section el el σK + d . In our notation, the amplitude MK + d reads dp el ˆ + (q2 )M ˆ KN Ψ ˆ 1 (q1 )} (15) = 2 Tr{Ψ MK +d 2 (2π)3 (q1 = p − P/2,
q2 = p − P1 /2),
ˆ 1,2 (q1,2 ) are the operators in the DWF (6) where Ψ of the initial and final deuteron, respectively, and ˆ K + N = 8π √sKN fˆK + N is the KN -scattering opM erator (the particle momenta are denoted in Fig. 3). The expression for the K + d → K + d differential cross section dσ/dΩ can be found, for example, in [23, 32] for the s-wave DWF and in [33] for the case with the d-wave part of DWF included. For the unpolarized particles, this cross section is dσ/dΩ = el |2 /(8π √s 2 |MK +d Kd ) , where Ω is the c.m. solid angle, and can be written as
4sKN dσ = (16) |Ap + An |2 Φ2S (q) + Φ2Q (q) dΩ sKd 2 2 2 + |Bp + Bn | ΦM (q) . 3 1 √ Here, q = (k − k1 ); sKd is the total c.m. energy; 2 Ap,n and Bp,n are the coefficients in the K + N amplitudes fˆp,n = Ap,n + Bp,n (nσ) (we calculate them at 1 fixed momentum p = (P1 + P2 ) of the intermediate 2 nucleon Fig. 3). The form factors ΦS (q), ΦQ (q), and ΦM (q) in Eq. (16) are Fd ΦQ = 2Fc − √ , 2 Fb Fc Fd +√ + , = Fa − 2 2 2
ΦS = Fa + Fb , ΦM where
(17)
Fa = 4π
drj0 (qr)u2 (r),
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p P
el MKd
P1
Fig. 3. A single-scattering diagram for the elastic process K + d → K + d. The lines of the particles are the same as in Fig. 2.
Fb = 4π drj0 (qr)w2 (r), Fc = 4π drj2 (qr)u(r)w(r), Fd = 4π drj2 (qr)w2 (r). Here, u(r) and w(r) are the s- and d-wave components of the DWF representation,
in2 coordinate 2 normalized as 4π dr u (r) + w (r) = 1; j0 and j2 are the zeroth- and second-order spherical Bessel functions, respectively. In the case of the Bonn [28] or Paris [29, 30] DWF, the analytical expressions for the integrals (18) are given in Section 5 of Appendix. 4. CROSS SECTONS OF Kd REACTIONS
4.1. The Results of Calculations and Comparison with the Data Here, we present the results of our calculations of the K + d cross sections based on the formulas from Sections 2 and 3. We use the KN phase shifts from [8] and take into account only s- and p-wave KN amplitudes, while d- and f -wave contributions are negligibly small at the energies of interest and are neglected. We also use the DWF of the Bonn potential [28] (full model) with s- and d-wave components included. Figure 4 shows the total cross sections of the reactions K + d → K + pn (a) and K + d → K 0 pp (b). The symbols correspond to the experimental data from [21] (Fig. 4a) and [21, 34–36] (Fig. 4b). The curves show the results of calculations. In Fig. 4a, the curves show the contributions of the amplitudes M1 + M2 (dashed), MR (dotted), M1 + M2 + MR (solid), M1 + M2 + MR with the s-wave DWF (dashdotted), and M1 + M2 + MR with the s-wave KN amplitudes (dashed curve “S”). In Fig. 4b, the curves show the contributions of the amplitude M1 + M2 . Here are also given the results obtained with s-wave
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σ, mb 25 K+d → K+pn, 20
(a)
[21]
15 S 10
5
0 15 K+d → K 0pp
10
(b)
[21] [34] [35] [36]
5
S 0
200
400
600
800 plab, MeV
Fig. 4. The total cross sections of the reactons (a) K + d → K + pn and (b) K + d → K 0 pp. The symbols correspond to the experimental data (see references on the plot). The curves present the results of computations. Solid curves show the contributions of the amplitudes M1 + M2 + MR (Fig. 4a) and M1 + M2 (Fig. 4b); dash-dotted curves are the results obtained with the s-wave part of DWF; dashed curves “S” are the results obtained with the s-wave KN amplitudes. The dashed and dotted curves in Fig. 4a show the contributions of the amplitudes M1 + M2 and MR , respectively.
DWF (dash-dotted curve) and with the s-wave KN amplitude (dashed curve “S”). Comparing the solid and dash-dotted curves in Fig. 4, one finds that the influence of the deuteron d-wave on the results is very small. Figure 4a shows that the contribution of the N N FSI amplitude MR substantially affects the calculated cross section. The term MR destructively interferes with the pole amplitudes M1,2 and decreases
the cross section at plab < 200 MeV/c by several times. Remember that the diagram MR , in which we take into account only the s-wave KN scattering, contains the N N rescattering only in the triplet 3S
1
state (SN N = 1, I = 0). However, the N N -scat-
tering length in the singlet 1 S0 state (SN N = 0, I = (1)
1) aN N = 24 fm [27] is large in comparison with the PHYSICS OF ATOMIC NUCLEI Vol. 71 No. 8
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σ, mb 50 (a)
K+d (total) 40
30
20
10
[10, 11] [12] [13] [21] [34]
0 50 K+d → K+d
(b)
40
30
20
10
0
[32] [37] 200
400
600
800 plab, MeV
Fig. 5. The K + d (a) total and (b) elastic cross sections. The symbols correspond to the experimental data (see references on the plot). The curves show the results of computations. The total K + d cross section (solid curve in Fig. 5a) is a sum of cross sections, shown by solid curves in Figs. 4a, 4b, and 5b. The dashed curve in Fig. 5a corresponds to the same sum, but with contribution of the K + d → K + pn cross section (dashed curve in Fig. 4a), not corrected for N N FSI. The dash-dotted curves show the results obtained with the s-wave part of DWF. (0)
triplet value aN N [see (13)]. Thus, one needs arguments to neglect the N N (1 S0 ) rescattering. This approximation should be reasonable in the momentum range where the p-wave KN amplitude is small. Comparing the solid curve and the dashed one, marked by “S”, in Fig. 4a, we find the influence of the p-wave KN scattering in the reaction K + d → K + pn to be small at plab < 250 MeV/c. From this indirect estimation, we expect that the N N (1 S0 )-rescattering PHYSICS OF ATOMIC NUCLEI Vol. 71
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correction is not significant in this range, but this approximation is less reliable at larger momenta plab . In more accurate calculation of the N N FSI correction the p-wave KN amplitude and N N (1 S0 ) interaction should also be included. We postpone this to a future study. t Figures 5a and 5b show the total (σK + d ) and el + the elastic (σK + d ) K d cross sections, respectively. The symbols are the experimental data from [10–
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σ, mb 20 K+d – K+pn
σ, mb 0.6 (a)
K+d – K 0pp
(b)
1 0.4 2 10 1 0.2 2 1 2 0 60
0 60
K+d (total)
(c)
K+d → K+d
(d)
1
40
40
2 1
1 2 2 20 –0.2
–0.1
0 a0, fm
20 –0.2
–0.1
0 a0, fm
Fig. 6. Predictions for the K + d cross sections at the beam momentum plab = 127 MeV/c as functions of the isoscalar KN -scattering length a0 . The plots show the (a) K + d → K + pn, (b) K + d → K 0 pp, (c) K + d total, and (d) elastic cross sections. Curves 1 and 2 show the results for the values (taken from the table) a1 = −0.328 and a1 = −0.308 fm, respectively. The solid (dashed) curves in Figs. 6a and 6c show the results obtained with (without) N N FSI correction to the K + d → K + pn cross section.
12, 21, 34] (Fig. 5a) and [21, 37] (Fig. 5b). Here, t the calculated cross section σK + d is taken as a sum + + + of the K d → K pn, K d → K 0 pp, and K + d → K + d cross sections, shown by the solid curves in Figs. 4a, 4b, and 5b. The dash-dotted curves in Fig. 5 are the results obtained with the s-wave DWF. The t dashed curve in Fig. 5a shows the result for σK +d in which the contribution of the K + d → K + pn cross section is not corrected for N N FSI. Let us comment here on the results of [5], where t data on the total cross section σK + d were analyzed t and σK + d was evaluated through the unitarity with single- and double-scattering K + d → K + d amplitudes employed. The unitarity cuts of the single- and double-scattering terms correspond to the contribu-
tions (summed over the K + pn and K 0 pp channels) of the squares |M1 |2 + |M2 |2 and of the interference term 2Re(M1∗ M2 ), where M1,2 are the pole amplitudes. Our comment is the following: t (1) The cross section σK + d calculated in [5] does not include the contribution of the elastic K + d scattering, which dominates at low momenta plab < 100 MeV/c. Results of our computations at plab = 100 MeV/c are shown in Figs. 4 and 5. They el inel + 0 give σK + d = 34.4 mb, σK + d = σ(K pn + K pp) = t 0.9 mb (9.7 mb), and σK + d = 35.3 mb (44.1 mb) with (without) the N N FSI diagram MR included. On the other hand, in [5] (see Fig. 5 there), one finds t inel inel σK + d = σK + d ≈ 27 mb. This value of σK + d is too large because of the following reasons. Firstly, the
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elementary KN amplitude used in [5] corresponds to the real (not virtual) target nucleon [see Eq. (17) there]. Secondly, they average the K + N cross section over the Fermi momentum distribution [see Eq. (19) there] in the range 0 < p < ∞, neglecting the kinematical boundaries. Thus, the cross section in [5] is overestimated in the near-threshold region, where the kinematical boundaries are important. (2) The double-scattering K + d amplitude is considered in [5] under the following assumptions. The propagator of the intermediate kaon is taken in a “static-nucleon” approximation. Thus, the contribut tion of this diagram to σK + d depends on the energy like the two-particle phase space instead of the threeparticle (KN N ) one as it should be. The elementary K + N amplitudes modified by the K + N (I = 0)resonance contribution are taken out of the integral over the momenta in the intermediate KN N state and taken at fixed nucleon momenta. This approximation is widely used for the hadronic amplitudes, usually being smooth functions in comparison with the rapid p dependence of the nuclear wave functions. However, in the case of a narrow (Γ ∼ 1 MeV) K + N resonance, this approximation can be unreliable. Summarizing the results of this section, we conclude that the approach based on the pole diagrams and modified by the N N FSI term (simplified as discussed above) gives a reasonable description of the existing data on the integrated K + d cross sections in the range plab < 800 MeV/c. The N N FSI effect is found to be large. It would be useful to have data t + + on the σK + d and K d → K pn cross section in the range, say, plab < 400 MeV/c (where they are absent now) for more detailed study of the N N FSI effect and comparison with the data.
4.2. On the Extraction of the Isoscalar KN (I = 0)-Scattering Length To determine the isoscalar KN -scattering length a0 , one needs additional data on kaon–neutron scattering, but neutron targets do not exist. Thus, to extract the value of a0 , one should compare the theoretical predictions with the data on the cross sections for the existing targets and the deuteron one is preferable. As a source of slow kaons, the decay φ(1020) → K + K − at rest can be used and φ(1020) mesons can be produced in e+ e− collisions at the DAΦNE accelerator in Frascatti. Figure 6 shows our predictions for the K + d cross sections at the initial momentum plab = 127 MeV/c, which is the kaon momentum in the φ(1020) decay. At this momentum, the p-wave KN amplitudes are negligibly small and we use here only the s-wave amplitudes. Figures 6a and 6b show the K + d → K + pn PHYSICS OF ATOMIC NUCLEI Vol. 71
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and K + d → K 0 pp cross sections. The total and the elastic K + d cross sections are given in Figs. 6c and 6d, respectively. The results are presented as functions of a0 in some range around the “realistic” values given in the table. The results are given for two fixed values a1 = −0.328 fm (curves 1) and a1 = −0.308 fm (curves 2), taken from the table. The solid and dashed curves in Figs. 6a and 6c show the results obtained with and without N N FSI taken into account. Thus, the N N FSI amplitude strongly affects the K + d → K + pn and total K + d cross sections for slow kaons. 5. CONCLUSIONS The theoretical predictions for the K + d cross sections were presented in the “quasi-elastic” energy range plab < 0.8 GeV/c, where the particleproduction processes in the elementary KN interactions can be neglected. We used the approach which employs the pole Kd → KN N amplitudes, the N N FSI correction, and the “realistic” KN phase shifts. In our approximation, we neglected the p-wave KN scattering when calculating the N N FSI correction. Since the s-wave KN -scattering amplitude is non-spin-flip, the N N FSI takes place only in the N N (3 S1 ) state, forbidden for the reaction K + d → K 0 pp because of the Pauli principle. This approximation should be reasonable in the lowmomentum range (estimated in Section 4), where the KN -scattering amplitude is predominantly s wave. A reasonable description of the data on the integrated K + d → K + pn and K + d → K 0 pp cross sections as well as on the total and elastic K + d cross sections was obtained. The N N FSI diagram affects strongly the value of the K + d → K + pn cross section in the low-energy region and interferes destructively with the pole diagrams. However, data on the K + d → K + pn cross section are available only at higher energies plab > 600 MeV/c, where our calculations of N N FSI are less reliable and some defects in the theoretical description are seen. It would be interesting to measure the K + d → K + pn cross section at low enegies, say, plab < 400 MeV/c, where the N N FSI effect is large, to investigate the role of this mechanism. Predictions were also given for the integrated cross sections of the K + d reactions with slow kaons as functions of the isoscalar KN -scattering length a0 . These results would be useful for extraction of the a0 value from the data. The corresponding experiments with a slow kaon beam from the φ(1020) decays may be proposed, say, for the DAΦNE accelerator. At this energy (plab = 127 MeV/c), the N N FSI effect is very strong in the reaction K + d →
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K + pn, as is seen from Fig. 4a. Thus, study of this reaction at the DAΦNE machine would be very important. In a more detailed treatment of N N FSI, the p-wave KN scattering and N N (1 S0 ) interaction should also be taken into account. We postpone this for a future study. However, these improvements should not significantly affect our results for the K + d cross sections at the energy of kaons from φ(1020) decay and the conclusion about the large magnitude of the N N FSI effect. ACKNOWLEDGMENTS We acknowledge support from the Federal Agency of Atomic Energy of the Russian Federation. Participation of V.E.T. and A.E.K. was supported by DFG–RFBR grant no. 05-02-04012 (436 RUS 113/820/0-1(R)). APPENDIX 1. CROSS SECTIONS AND PHASE SPACES Calculating the Kd cross sections, we use the invariant amplitudes with Feynman normalization. The cross section σn of the process a + b → 1 + . . . + n reads 1 √ |M |2 dτn , (A.1) σn = 4qab s n d3 pi . dτn = In (2π)4 δ(4) (Pi − Pf ) (2π)3 · 2Ei i=1 √ Here, M is the invariant amplitude; s is the total c.m. energy; qab is the initial relative momentum, where λ(z, x, y) = qab = λ(s, m2a , m2b ), √ 2 (z − x − y) − 4xy/2 z and ma (mb ) is the mass of the particle a(b); dτn is the element of the final nparticle phase space; Pi (Pf ) is the total initial (final) four-momentum; Ei and pi are the total energy and momentum of the ith final particle; the factor In ≡ 1/n1 ! . . . nk ! takes into account the identity of final particles, where ni is the number of particles of the ith type (n1 + . . . + nk = n). Then, the cross section σ of the reaction Kd → KN N with unpolarized particles can be written as √ dσ d sN N , (A.2) σ= √ d sN N In dσ = |M |2 Q1 pdz1 dzdϕ, √ d sN N 2(4π)4 Qs where z = cos θ and z1 = cos θ1 . Here, |M |2 is the square of the reaction amplitude M with unpolarized
particles; Q = |Q| and Q1 = |Q1 |, where Q and Q1 are the c.m. momenta of the incoming and outgoing kaon, respectively; θ1 is the c.m. polar angle of the outgoing kaon; p = |p|; p, θ, and ϕ are the momentum and the angles (polar and azimuthal) of the outgoing nucleon, say N1 , in the N N rest frame. 2. KINEMATICS Let us express all the momenta used to calculate the √ Kd → KN N amplitude through the variables sN N , z1 , z, and ϕ of integrals (A.2). One can write Q = λ(s, m2K , m2d ), Q1 = λ(s, m2K , sN N ), (A.3) √ p = m( sN N − 2m), where mK (md ) is the kaon (deuteron) mass and the function λ(. . .) is defined after Eq. (A.1). Let us introduce the following notation: p1,2 and p1,2 are the momenta of the final nucleons in the reaction and in the deuteron rest frames, respectively; q1,2 and q1,2 are the initial and final nucleon momenta, respectively, in the rest frame of the KN1,2 → KN1,2 subprocess in the diagram M1,2 (Fig. 2). Then we write Q1 Q , p1,2 = p1,2 + , (A.4) p1,2 = ±p − 2 2 ωp1,2 − mQ1 , q1,2 = q1,2 + Q1 − Q, q1,2 = m+ω where ω = m2K + p2lab is the kaon total energy.
The values qi = |qi |, zKNi = (qi qi )/qi qi , and ni = [qi × qi ]/|[qi × qi ]| are used to calculate the K + Ni scattering amplitude (i = 1, 2), according to (3). 3. THE SQUARE OF THE Kd → KN N AMPLITUDE Here, we write out the square |M |2 of the amplitude M = M1 + M2 + MR for unpolarized particles, applying the formulas from Sections 2 and 3. Hereafter, we exclude the isospin variables and fix the nucleons with momenta p1 and p2 in the reaction K + d → K + pn as proton and neutron, respectively. Then, In = 1(1/2) in Eq. (A.2) for the reaction K + d → K + pn (K + d → K 0 pp). Let A(i) and B(i) be the coefficients in the K + Ni amplitude fˆKNi = A(i) + B(i) (ni σ). Then A(1) = A1 and B(1) = B1 1 1 (A(2) = (A1 + A0 ) and B(2) = (B1 + B0 )) for the 2 2 K + p (K + n)-scattering subprocess in the diagram M1 (M2 ) for the reaction K + d → K + pn; A(i) = PHYSICS OF ATOMIC NUCLEI Vol. 71 No. 8
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1 − A0 ) and B(i) = (B1 − B0 ) for the charge 2 exchange subprocess K + n → K 0 p in the reaction K + d → K 0 pp; the values AI and BI are given in (3). For the different terms of the expression |M |2 , we obtain (hereafter Tr{(. . .)} means the trace of the 2 × 2 matrix (. . .) with spin indices) ˆ j )Ψ ˆ + (pj )fˆ+ fˆKN } |Mi |2 = c2 Tr{Ψ(p (A.5)
gj Re(Mi+ MR ) = c2 2Re AR A∗(i) fj + A 3
3(pj ∆)2 gj ∗ −1 + Im(AR B(i) B) + B 3 p2j ∆2 (pj ∆) (i = j = 1, 2), × gj 2 2 (pi [ni × pj ]) pj ∆
(i = j = 1, 2),
|MR |2 = 2c2 |AR |2 (|A|2 + 2|B|2 ),
ˆ 2 )(A(2) − B(2) n ˆ + (p1 ) M1 M2+ = ±c2 Tr{Ψ(p ˆ 2 )+ Ψ
where f1,2 = f (p1,2 ), g1,2 = g(p1,2 ). Here, A and B ˆ are the coefficients in the expression for L(x), given below in the next section. If one neglects the d-wave part of DWF, Eqs. (A.7) give
1 2 (A1
KNi
i
ˆ 1 )}, × (A(1) + B(1) n ˆ + (pj )fˆ+ L} ˆ Mi+ MR = c2 AR Tr{Ψ KNi ˆL ˆ + }, |MR |2 = c2 |AR |2 Tr{L
(i = j = 1, 2),
2 ˆ = L(−p ˆ ˆ 2 ), L ) − L(β
where n ˆ i = (ni σ). Here, the sign “−” in the expression for M1 M2+ corresponds to the case of the reaction K + d → K 0 pp; the terms Mi+ MR and |MR |2 are given for the reaction K + d → K + pn; the quantities ˆ are given by (14); calculating the factors AR and L(x) ci and c, defined in Eqs. (8) and (14), we use the value √ of sKN for the nucleon at rest, i.e., ci = c = 16π msKN and sKN = m2K + m2 + 2ωm. Let us introduce the functions f (p) and g(p), ˆ rewriting Ψ(p) (6) in the form (p)(σ) ˆ , (A.6) Ψ(p) = f (p)(σ) + g(p) p2 3w(p) u(p) w(p) , g(p) = − . f (p) = √ + 2 2 2 Finally, from Eqs. (A.5) and (A.6), we obtain |Mi |2 = c2 [u2 (pj ) + w2 (pj )]
(A.7)
× [|A(i) |2 + |B(i) |2 ] (i = j = 1, 2),
+ 2 ∗ Re(M1 M2 ) = ±c Re(A(1) A(2) ) u(p2 )u(p1 )
3zp2 − 1 + w(p2 )w(p1 ) 2
+
2g1 g2 zp 3p1 p2
∗ × (p2 [n2 × p1 ])Im(A(1) B(2) ) − (p1 [n1 × p2 ]) 2 ∗ ) (n1 n2 ) × Im(B(1) A∗(2) ) + Re(B(1) B(2) 3 × f1 f2 + g1 f2 + f1 g2 + g1 g2 zp2
4(p1 n1 )(p2 n2 ) g1 f2 f1 g2 g1 g2 zp + 2 − + 3 p1 p2 p21 p2 (p1 p2 ) , zp = p1 p2
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|Mi |2 = c2 u2 (pj )[|A(i) |2 + |B(i) |2 ]
(A.8)
(i = j = 1, 2),
M1 M2+ = ±c2 u(p1 )u(p2 ) A(1) A∗(2) 1 ∗ + B(1) B(2) (n1 n2 ) , 3 + Mi MR = c2 AR A∗(i) Au(pj ), |MR |2 = 2c2 |AR |2 |A|2 . ˆ 4. CALCULATION OF THE OPERATOR L(x) Here, it is convenient to use the DWF in coordinate representation, i.e., w(r) u(r) ˆ (A.9) Φ(r) = √ (σ) − 2r r 2 3(r)(rσ) ˆ ˆ − (σ) , Ψ(q) = d3 re−iqr Φ(r), × r2 ˆ where Ψ(q) is given by (6). For the DWF of the Bonn [28] and Paris [29] potentials, the s- and d-wave functions (u and w, respectively) were parametrized [28, 30] in the form Ci , (A.10) u(p) = p2 + m2i i Di w(p) = p2 + m2i i Di Ci = Di = Di m2i = =0 , m2i i i i i Ci e−mi r , 4π i
Di 3 3 −mi r e + 1+ . w(r) = 4π mi r m2i r 2 i u(r) =
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ˆ Calculating the integral L(x) (14), we transform 2 ˆ the factors Ψ(q + ∆) and (q + x − 0)−1 into the r representation and obtain 3 d r i∆r+αr ˆ ˆ e Φ(r), (A.11) L(x) = 2 r −a (x > 0), α= a = |x|. ia (x < 0),
Fa,b,c,d (18) analytically, and we obtain Ci Cj (A.14) Fa = Aij 4π∆ ij
∆ Aij ≡ arctan , mi + mj Di Dj 3(x + y + ∆2 )2 − 4xy Aij Fb = 8π 4xy∆ ij
ˆ ˆ Let us rewrite the operator Φ(r) in (A.9) as Φ(r) = Φij εi σj , where εi is the ith component of the deuteron polarization vector. Then, from Eq. (A.11), we arrive at the expressions ˆ L(x) = Lij εi σj ,
3∆i ∆j − δij , Lij = Aδij + B ∆2
(A.12)
√ d3 r i∆r+αr e u(r) A= 2 r2 √ = 4 2π dreαr u(r)j0 (r∆),
3 d r i∆r+αr 3(∆r)2 1 e w(r) − 1 B=− 2 r2 r 2 ∆2 = 4π dreαr w(r)j2 (r∆). Making use of the functions u(r) and w(r) (A.10), we obtain √ 2Ci J(mi , a, ∆), a = |x|, (A.13) A= i
B=
i
+
3(∆2
Di
(x = m2i , y = m2j ), Ci Dj 3x FC = − 8π 4mj ∆2 ij 4y∆2 + 3(x − y + ∆2 )2 + Aij , 4y∆3 Di Dj Fd = 8π
3(∆2 + x − m2i ) 8mi ∆2
+ x − m2i )2 + 4m2i ∆2 J(mi , a, ∆) , 8m2i ∆2
where ∆ 1 arctan (x > 0), ∆ m+a a+∆ a−∆ 1 arctan − arctan J(m, a, ∆) = 2∆ m m 2 2 i m + (a + ∆) (x < 0). + ln 2 2 m + (a − ∆)2 J(m, a, ∆) =
ij
×
3(x + y +
− (x − y)2 ] − 8xy∆2 Aij . 8xy∆3
∆2 )[∆4
6. ESTIMATION OF THE COULOMB CROSS SECTION σC Here, we estimate the pure Coulomb cross section of the reaction K + d → K + pn in the nonrelativistic case. With the s-wave DWF u(p), we can write dp 2 u (p) dΩ|fC |2 , (A.15) σC = (2π)3 2αc µ . fC = t Here, p is the neutron-spectator three-momentum in the deuteron rest frame; fC is the Born amplitude of the Coulomb K + p scattering; dΩ = dzdϕ is the solid angle element in the final K + p system; αc ≈ 1/137; µ = mK m/(m + mK ) is the reduced mass; and t is the four-momentum transfer squared. In the nonrelativistic form, t = −(q1 − q)2 , where q1 (q) is the initial (final) relative three-momentum in the subprocess K + p1 → K + p and p1 is the virtual proton with the mass m1 = m. For the angular part of the integral dΩ|fC |2 , making use of the relations q12 − q 2 = 2µ(m − m1 ), p2 + α2 , α2 = mεd , m where εd is the deuteron binding energy, we obtain 4πα2 m2 (A.16) dΩ|fC |2 = 2 c 2 2 . (p + α ) m1 = m −
5. EXPRESSIONS FOR THE INTEGRALS Fa,b,c,d (18) For the wave functions u(r) and w(r), given by Eqs. (A.10), one can calculate the integrals
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We shall estimate the cross section σC√(A.15) with 2 the DWF of the simplest form u(p) = 8πα/(p + 2 α ). Formally, the integral dp in (A.15) depends on the kinematical boundaries through the condition √ sK + p > m + mK , but we shall calculate it in the range 0 < p < ∞. In this approximation, it is supposed that the DWF is a rapid function of p and the process K + d → K + pn is considered in the region not very close to the threshold. However, we take into account the p-dependent factor (A.16). Finally, we obtain ∞ p2 dp (A.17) σC = 16α2c αm2 (p2 + α2 )4 0
πα2 = 2c ≈ 6.5 mb. 2εd This value is not very small in comparison with the hadronic K + d → K + pn cross section, shown in Fig. 4a. We neglect the Coulomb contribution, since it is concentrated in the region of small scattering angles, which may be not accepted by the detectors. REFERENCES 1. T. Nakano et al. (LEPS Collab.), Phys. Rev. Lett. 91, 012002 (2003); V. V. Barmin et al. (DIANA Collab.), Phys. At. Nucl. 66, 1715 (2003); S. Stepanyan et al. (CLAS Collab.), Phys. Rev. Lett. 91, 252001 (2003); M. Danilov and R. Mizuk, hepex/704.3531v1 (2007). 2. R. A. Arndt, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C 68, 042201 (2003); 69, 019901 (2004); nucl-th/0308012. 3. J. Haidenbauer and G. Krein, Phys. Rev. C 68, 052201 (2003); hep-ph/0309243. 4. S. Nussinov, hep-ph/0307357. 5. W. R. Gibbs, Phys. Rev. C 70, 045208 (2004); nuclth/0405024.
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6. R. N. Cahn and G. H. Trilling, Phys. Rev. D 69, 011501 (2004); hep-ph/0311245. 7. A. Sibirtsev, J. Haidenbauer, S. Krewald, and Ulf-G. Meißner, Phys. Lett. B 599, 230 (2004); hepph/0405099. 8. W. R. Gibbs and R. Arceo, nucl-th/0611095. 9. A. Sibirtsev, J. Haidenbauer, S. Krewald, and Ulf-G. Meißner, nucl-th/0608028. 10. T. Bowen et al., Phys. Rev. D 2, 2599 (1970). 11. T. Bowen et al., Phys. Rev. D 7, 22 (1973). 12. D. V. Bugg et al., Phys. Rev. 168, 1466 (1968). 13. A. S. Carrol et al., Phys. Lett. B 45, 531 (1973). 14. R. A. Burnstein et al., Phys. Rev. D 10, 2767 (1974). 15. T. F. Kycia et al., Phys. Rev. 118, 553 (1960). 16. C. J. Adams et al., Phys. Rev. D 4, 2637 (1971). 17. C. J. Adams et al., Nucl. Phys. B 66, 36 (1973). 18. W. Cameron et al., Nucl. Phys. B 78, 93 (1974). 19. V. Cook et al., Phys. Rev. Lett. 7, 182 (1961). 20. R. L. Cool et al., Phys. Rev. D 1, 1887 (1970). 21. G. Giacomelli et al., Nucl. Phys. B 37, 577 (1972). 22. V. E. Tarasov, V. V. Baru, and A. E. Kudryavtsev, Yad. Fiz. 63, 871 (2000) [Phys. At. Nucl. 63, 801 (2000)]. 23. V. J. Stenger et al., Phys. Rev. 134, B1111 (1964). 24. A. B. Migdal, Sov. Phys. JETP 1, 2 (1955). 25. K. M. Watson, Phys. Rev. 88, 1163 (1952). 26. V. M. Kolybasov, V. G. Ksenzov, Sov. J. Nucl. Phys. 22, 372 (1975). 27. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: NonRelativistic Theory (Nauka, Moscow, 1989; Pergamon, New York, 1977), p. 641. 28. R. Machleidt et al., Phys. Rep. 149, 1 (1987). 29. M. Lacombe et al., Phys. Rev. C 21, 861 (1980). 30. M. Lacombe et al., Phys. Lett. B 101, 139 (1981). 31. S. S. Kamalov, J. A. Oller, E. Oset, and M. J. VicenteVacas, Phys. Rev. C 55, 2985 (1997). 32. G. Giacomelli et al., Nucl. Phys. B 68, 285 (1974). 33. R. G. Glasser et al., Phys. Rev. D 15, 1200 (1977). 34. C. J.S. Damerell et al., Nucl. Phys. B 94, 374 (1975). 35. M. Sakitt et al., Phys. Rev. D 15, 1846 (1977). 36. W. Slater et al., Phys. Rev. Lett. 7, 378 (1961). 37. M. Sakitt et al., Phys. Rev. D 12, 3386 (1975).
