Third order Bose-Einstein correlations by means of Coulomb wave function revisited
Minoru Biyajima1 ? Takuya Mizoguchi2 ? Naomichi Suzuki3 ? 1 Department of Physics, Shinshu University, Matsumoto, 390-8621, Japan 2 Toba National College of Maritime Technology, Toba 517-8501, Japan 3 Department of Comprehensive Management, Matsumoto University, Matsumoto 390-1295, Japan
arXiv:hep-ph/0512275v2 22 Mar 2006
Abstract In previous works, in order to include correction by the Coulomb wave function in Bose-Einstein correlations (BEC), the two-body Coulomb scattering wave functions have been utilized in the formulation of three-body BEC. However, the threebody Coulomb scattering wave function, which satis?es approximately the three-body Coulomb scattering Schr¨ odinger equation, cannot be written by the product of the two-body scattering wave functions. Therefore, we reformulate the three-body BEC, and reanalyze the data. A set of reasonable parameters is obtained.
Recently, in addition to the data on the two-body charged Bose-Einstein correlationsiBEC), data on the three-body charged BEC have been reported [1, 2, 3]. In some papers [1, 3], the Coulomb correction is done with ?xed source radius, for example, 5 fm. On the other hand, the quasi-corrected data (raw data with acceptance correction) on the two-body (2π ? ) BEC  and the three-body (3π ? ) BEC have been reported . In Ref. [6, 7], authors proposed a theoretical formula for the 3π ? BEC by the use of the two-body Coulomb wave functions, and outputted information on BEC with ?xed source radii (5 fm and 10 fm). On the other hand, we have analyzed the 2π ? and 3π ? BEC, using the CERN-MINUIT program with the two-body Coulomb wave functions and the source radius as a free parameter [8, 9, 10]. The formula for 2π ? BEC reduces to that of plane wave formulation in the limit of plane wave approximation. However, the formula for 3π ? BEC does not reduce to that of plane wave formulation [11, 12]. Additional factor (3/2) appears in the phase of plane wave [8, 9, 10]. Therefore, we have re-interpreted the source radius estimated from the analysis of 3π ? BEC.
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In this paper, we would like to examine the relation between the two-body Coulomb wave function and the asymptotic solution of the three-body Coulomb wave function, which cannot be written by the product of two-body Coulomb wave functions. In addition, we would like to show that factor 3/2 disappears from the phase factors of plane wave in the formulation of the 3π ? BEC, if the correct asymptotic three-body Coulomb wave function is used. In the second section, an asymptotic solution for the three-body Coulomb scattering Sch¨ odinger equation is shown. The formula for 3π ? BEC is derived from the analogy of the formula for the plane wave formulation in the third section. Analysis of 3π ? BEC is done in the fourth section. Final section is devoted to summary and discussions.
Approximate solution for Sch¨ odinger equation of threebody Coulomb scattering
In order to describe the two-body charged BEC ( for example, 2π ? system ), we should solve the Shr¨ odinger equation of Coulomb scattering. The solution, which is regular at the origin of the Coulomb potential, is given by
C (xi , xj ) = eikij ·rij φkij (rij ), ψk i kj
φkij (rij ) = Γ(1 + iηij )eπηij /2 F [?iηij , 1; i(|kij | · |rij | ? kij · rij )],
for particles i and j , where the coordinate and momentum of particle i, are denoted by xi and ki , respectively, and ei in ηij is the charge of particle i. In Eq. (1), the relative coordinate and momentum of particles i and j are denoted by rij = xi ? xj , and kij = (mj ki ? mi kj )/(mi + mj ), respectively, ηij = ei ej ?ij /|kij | where ?ij is reduced mass of mi and mj , F [a, b; x] is the con?uent hypergeometric function, and Γ(x) is the Gamma function. In order to describe the three-body Coulomb scattering, the Jacobi coordinates  are introduced; ζ1 = x2 ? x1 , ζ2 = x3 ? (m1 x1 + m2 x2 )/M2 ,
ζ3 = (m1 x1 + m2 x2 + m3 x3 )/M, M = m1 + m2 + m3 .
M2 = m1 + m2 , The relative coordinates are written as,
r32 = x3 ? x2 = ?β ζ1 + ζ2 , α = m2 /M2 , β = m1 /M2 . The Schr¨ odinger equation of the three-body Coulomb scattering is given by, ? 1 2 e1 e2 e2 e3 e3 e1 P2 P2 1 2 ?ζ1 ? ?ζ2 + + + ? 1 ? 2 Ψf = 0, 2?1 2?2 r12 r23 r31 2?1 2?2 2 (3)
r31 = x3 ? x1 = αζ1 + ζ2 ,
r21 = x2 ? x1 = ζ1 ,
Figure 1: Jacobi coordinates of three-body system. The starting point of ζ2 is the center of mass of particles 1 and 2.
where ?1 = m1 m2 /M2 , ?2 = M2 m3 /M, P1 = ?1 dζ1 /dt = (m1 k2 ? m2 k1 )/M2 ,
P2 = ?2 dζ2 /dt = (M2 k3 ? m3 (k1 + k2 ))/M.
Then, the approximate solution for the Schr¨ odinger equation in ?0 , where r12 , r23 , r31 >> 1, is given by [14, 15], Ψf = ei(P1 ζ1 +P2 ζ2 ) φk12 (r12 )φk23 (r23 )φk31 (r31 ). The phase factor of the plane wave in Eq.(4) is rewritten as, m2 + m3 m3 + m1 m1 + m2 k12 · r12 + k23 · r23 + k31 · r31 , P1 ζ1 + P2 ζ2 = M M M 2 = (k12 · r12 + k23 · r23 + k31 · r31 ), 3 where m1 = m2 = m3 is used. Therefore, the solution Ψf is written as , Ψf
C′ C C C = ψk (x1 , x2 )ψk (x2 , x3 )ψk (x3 , x1 ), 1 k2 2 k3 3 k1
′ ′ ′
ψki kj (xi , xj ) = ei(2/3)kij rij φkij (rij ).
The approximate solution for the three-body Coulomb scattering can be written by the C ′ (x , x ), but not the product of the wave function of two-body scattering, product of ψk i j i kj C (x , x ). In the correct formula, factor 2/3 is multiplied to the phase of plane wave. ψk i j i kj In Ref., the Coulomb correction for n-body scattering is discussed, where the nbody Coulomb scattering wave function is given by the product of two-body Coulomb scattering wave functions. However, the n-body Coulomb scattering wave function in ?0 is approximately given by
C′ ψk (xi , xj ) i kj
C (xi , xj ), ψk i kj
= ei(2/n)kij rij φkij (rij )
for n ≥ 3.
formula for third order BEC
The wave function of identical Bose particles should be symmetrized. In Fig.2, or Fig.3, Vc denotes the interaction between two particles by the Coulomb potential, and cross (X) represents the exchange of particles. As is shown in Fig.2, the two particle momentum density is given by, N (2π
1 2 2 i=1
C C ρ(xi )d3 xi |ψk (x1 , x2 ) + ψk (x2 , x1 )|2 1 k2 1 k2
ρ(xi )d3 xi (G1 + G2 ),
G1 = G2 where
1 C C |ψk (x1 , x2 )|2 + |ψk (x2 , x1 )|2 , 1 k2 1 k2 2 C C? = Re ψk (x1 , x2 )ψk (x2 , x1 ) , 1 k2 1 k2
1 x2 ]. exp[ ? 2R2 (2πR2 )3/2
Figure 2: Two-body BEC diagram The exchange diagram for 3π ? BEC is shown in Fig.3. According to the diagram, the three particle density for 3π ? BEC is written as, N (3π where
C C C A(1) = A1 = ψk (x1 , x2 )ψk (x2 , x3 )ψk (x3 , x1 ), 1 k2 2 k3 3 k1 C C C A(2) = A23 = ψk (x1 , x3 )ψk (x3 , x2 )ψk (x2 , x1 ), 1 k2 2 k3 3 k1 C C C A(3) = A12 = ψk (x2 , x1 )ψk (x1 , x3 )ψk (x3 , x2 ), 1 k2 2 k3 3 k1
′ ′ ′ ′ ′ ′ ′ ′ ′ ?)
1 3 6 i=1
ρ(xi )d3 xi |
A(j )|2 ,
C C C A(4) = A123 = ψk (x2 , x3 )ψk (x3 , x1 )ψk (x1 , x2 ), 1 k2 2 k3 3 k1 C C C A(5) = A132 = ψk (x3 , x1 )ψk (x1 , x2 )ψk (x2 , x3 ), 1 k2 2 k3 3 k1 C C C A(6) = A13 = ψk (x3 , x2 )ψk (x2 , x1 )ψk (x1 , x3 ). 1 k2 2 k3 3 k1
′ ′ ′ ′ ′ ′
1? Vc Vc 1
Vc 2 (a) 3 1
Vc 2 (b) 2? 2? Vc 3?
Vc 3 1
Vc 2 (c) 1? 3? Vc 1?
1? Vc Vc 1
Vc 2 (d) 3 1
Vc 2 (e)
Vc 3 1
Vc 2 (f)
Figure 3: Three-body BEC diagram
In the plane wave approximation, each amplitude A(i) approaches to the following form, A(1) = A1 ?→ ei(2/3)(k12 ·r12 +k23 ·r23 +k31 ·r31 ) = ei(k1 ·x1 +k2 ·x2 +k3 ·x3 ) ,
PW PW PW
A(6) = A13 ?→ ei(2/3)(k12 ·r32 +k23 ·r21 +k31 ·r13 ) = ei(k1 ·x3 +k2 ·x2 +k3 ·x1 ) .
A(5) = A132 ?→ ei(2/3)(k12 ·r31 +k23 ·r12 +k31 ·r23 ) = ei(k1 ·x3 +k2 ·x1 +k3 ·x2 ) ,
A(4) = A123 ?→ ei(2/3)(k12 ·r23 +k23 ·r31 +k31 ·r12 ) = ei(k1 ·x2 +k2 ·x3 +k3 ·x1 ) ,
A(3) = A12 ?→ ei(2/3)(k12 ·r21 +k23 ·r13 +k31 ·r32 ) = ei(k1 ·x2 +k2 ·x1 +k3 ·x3 ) ,
A(2) = A23 ?→ ei(2/3)(k12 ·r13 +k23 ·r32 +k31 ·r21 ) = ei(k1 ·x1 +k2 ·x3 +k3 ·x2 ) ,
In Eq.(10), PW means the plane wave approximation of the amplitude, and the condition in the center of mass system, exp[?i(k1 + k2 + k3 ) · ζ3 ] = 1 is used. The amplitudes squared in Eq.(8) can be classi?ed into the following groups, F1 = F12 = F23 = 1 ? ? ? ? ? [A1 A? 1 + A12 A12 + A23 A23 + A13 A13 + A123 A123 + A132 A132 ], 6 1 ? ? [A1 A? 12 + A23 A123 + A13 A132 + c.c.], 6 1 ? ? [A1 A? 23 + A12 A132 + A13 A123 + c.c.], 6 5
F31 = F123 = F132 =
1 ? ? [A1 A? 13 + A23 A132 + A12 A123 + c.c.], 6 1 ? ? ? ? ? [A1 A? 132 + A132 A123 + A13 A12 + A12 A23 + A23 A13 + A123 A1 ], 6 1 ? ? ? ? ? [A1 A? 123 + A23 A12 + A12 A13 + A123 A132 + A132 A1 + A13 A23 ], 6
where, c.c. denotes the complex conjugate, and F132 is the complex conjugate of F123 . In the plane wave approximation, F1 reduces to 1, Fij corresponds to exchange between i and j charged particles, and F123 correspond to exchange among three charged particles. Phenomenologically, the coherence parameter λ is introduced into the formula for 2π ? BEC as,
2 N 2π = C N BG i=1
ρ(xi )d3 xi (G1 + λG2 ),
where C is the normalization factor. In the third order BEC, factor λn/2 is multiplied to the amplitudes squared according to the number n of exchange particles. After ζ3 integration, the 3π ? BEC is given by N 3π N BG
ρ(xi )d3 xi F1 + 3λF12 + 2λ3/2 Re (F123 ) d3 ζ1 d3 ζ2 exp ? 1 2R2 1 2 2 2 ζ + ζ 2 1 3 2 F1 + 3λF12 + λ3/2 Re (F123 ) . (13)
C √ (2 3πR2 )3
The set of following variables is used in the concrete calculations of Eq.(13),
? ? ? k12 = ?P1 , ? ? ? ? ? 1 ?
? 1 ? k31 = 2 P1 + 3 ? 4 P2 , ? ? ? ? ? Q = 4(k2 + k2 + k2 ) = 3 12 23 31
k23 = 2 P1 ? 3 4 P2 ,
2 + 9P 2 . 6P1 2 2
Analysis of 3π ? BEC
The formula (13) is applied to the analysis of quasi-corrected data on 3π ? BEC by STAR Collaboration . The results are shown in Table 1 and Fig. 4. For comparison, the result of previous work  is also shown in the lower part of Table 1. The result for 2π ? BEC by STAR Collaboration  is shown in Table 2. The source radius R3rd estimated from the data on 3π ? BEC with Eq.(13) is comparable pre with that R2nd from 2π ? BEC. However, the source radius R3 rd estimated in the previous work , namely with the two-body Coulomb wave functions is much smaller than R3rd pre with Eq.(13). The re-interpreted radius becomes (3/2)R3 rd = 8.01 [fm], which is nearly equal to R3rd . The coherence parameter λ3rd estimated from 3π ? BEC with Eq.(13) is somewhat smaller than that λ2nd from 2π ? BEC.
Table 1: Analyses of 3π ? BEC by STAR Collaboration . Eq.(13) previous work  R [fm] 8.26±0.39 5.34±0.24 λ 0.50±0.02 0.56±0.02 χ2 /n.d.f . 1.88/35 2.80/35
Table 2: Analyses of 2π ? BEC by STAR Collaboration . Eq.(12)
2 1.8 N3π-/N BG 1.6 1.4 1.2 1 0.8 0 50 100 150 Q3 (MeV) 200
R [fm] 8.75±0.31
χ2 /n.d.f . 23.0/25
Figure 4: Analysis of quasi-corrected data on 3π ? BEC by STAR Collaboration  with Eq.(13).
The problem on the phase factors appearing in the two-body BEC among three identical particles [11, 12] is proposed in . If these factors are taken into account, the formula for 3π ? BEC is given by,
3 N 3π =C N BG i=1
ρ(xi )d3 xi F1 + 3λF12 + 2λ3/2 Re[F123 ] × W ,
where W = cos(φ12 + φ23 + φ31 ), which is parameterized as W = cos(g × Q3 ) in the simplest 2 2 2 form with parameter g and Q2 3 = (k1 ? k2 ) + (k2 ? k3 ) + (k3 ? k1 ) . The result is shown in Table 3 and Fig. 4. Estimated source radius shown in Table 3 is smaller than R3rd with Eq.(13), and is not consistent with R2nd . Table 3: Analyses of 3π ? BEC by STAR Collaboration with Eq.(15). Eq.(15) R [fm] 7.70±0.57 λ 0.55±0.05 g 31.13± 10.87 χ2 /n.d.f . 0.51/34
2 1.8 N3π-/N BG 1.6 1.4 1.2 1 0.8 0 50 100 150 Q3 (MeV) 200
Figure 5: Analysis of quasi-corrected data on 3π ? BEC by STAR Collaboration  with Eq.(15).
Summary and discussions
We reformulate the formula for 3π ? BEC, using the asymptotic three-body Coulomb wave function. We apply the formula to the analysis of data on 3π ? BEC by STAR Collaboration. The source radius R3rd estimated from 3π ? BEC is consistent with that R2nd from 2π ? BEC . The coherence parameter λ3rd estimated from 3π ? BEC with Eq.(6) is almost the same with λ2nd from 2π ? BEC. Whether a set of preferable parameters can be estimated from the analyses of 2π ? BEC and 3π ? BEC or not in other approaches will be reported elsewhere . By the use of our formula, we can estimate source radius with Coulomb correction, without the re-interpretation due to the factor (3/2).
One of the authors (N.S.) would like to thank J.R.Glauber for variable comments on the behaviors of charged particles in the electric ?eld at Kromeritz, August, 2005. They would like to thank E.O.Alt, T.Cs¨ org? o, and members of WA8 Collaboration ( Y.Miake, L.Rosselet) for discussing on this subject. They also would like to thank participants at a meeting of RCNP at Osaka University, Faculty of Science, Shinshu University, Toba National College of Maritime Technology and Matsumoto University for ?nancial support.
 H. B?ggild et al., NA44 Collaboration, Phys. Lett. B455(1999)77.  M.M.Aggarwal et al., WA98 Collaboration, Phys. Rev. C67(2003)014906  J.Adams et al., STAR Collaboration, Phys. Rev. Lett. 91(2003)262301  C.Adler, et al. STAR Collaboration, Rhys. Rev. Lett. 87(2001)082301
 R.Willson, Dr. Thesis at Ohio University (2002)  E.O.Alt, T.Cs¨ org? o, B.L¨ orstad and J.Schmidt-S?rensen, Phys. Lett. B458(1999)407  E.O.Alt, T.Cs¨ org? o, B.L¨ orstad and J.Schmidt-S?rensen, Eur. Phys. J. C13(2000)663  T. Mizoguchi and M. Biyajima, Phys. Lett. B499(2001) 245  M. Biyajima, M.Kaneyama and T. Mizoguchi, Phys. Lett. B601(2004)41  M. Biyajima, T. Mizoguchi and N.Suzuki, hep-ph/0510015, to be published in the proceedings of the Workshop on Particle Correlations and Femtoscopy, Kromeritz, Czech Republic, August 15-17, 2005  M. Biyajima, A. Bartl, et al., Prog. Theor. Phys.84(1990) 931; [addenda] Prog. Theor. Phys. 88(1992)157  N.Suzuki and M. Biyajima, 60(1999)034903 Prog. Theor. Phys.88(1992)609; Phys Rev. C
 M.Reed and B.Simon, Methods of modern mathematical physics V: Scattering theory, Academic press, 1979  E. O. Alt, Few-Body Systems Suppl., 10(1999)65-74; nucl-th/9809046  M.Brauner, J.S.Briggs and H.Klar, J. Phys.B22(1989)2265  N.Suzuki, K.Ide, M. Biyajima and T.Mizoguchi, Soryushiron Kenkyu(Kyoto), in Japanese, 112-4(2006)1  U.Heinz and Q.H.Zhang, Phys. Rev. C56(1997)426
pre  Previous result R3 rd = 1.53 ± 0.20 fm for S+Pb collision obtained in  becomes 4.40 ± 0.58 fm in re-analyses by the present formula Eq. (13). This is almost in agreement with the result R2nd = 4.69 ± 0.46 fm obtained therein .
 M. Biyajima, T. Mizoguchi and N.Suzuki, in preparation