¯ reaction Hadronic structure functions in the e+ e− → ΛΛ G¨oran F¨aldta , Andrzej Kupsca a Division
of Nuclear Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
arXiv:1702.07288v1 [hep-ph] 23 Feb 2017
Abstract ¯ Cross-section distributions are calculated for the reaction e+ e− → J/ψ → Λ(→ pπ ¯ − )Λ(→ pπ+ ), and the related reaction mediated by the ψ(2S ) meson. The hyperon-decay distribution depend on a number of structure functions that are bilinear in the, possibly complex, psionic form factors G E (P2 ) and G M (P2 ) of the Λ hyperon. The magnitudes and phases of these form factors can be ¯ hyperons. We illustrate how this method works by uniquely determined from the unpolarized joint-decay distributions of Λ and Λ analyzing some available experimental data. Keywords: PACS: 13.30.Eg Hadronic decays, 13.40.Gp Electromagnetic form factors, 13.66.Bc Hadron production in e− e+ interactions, 14.20.Jn Hyperons
1. Introduction Two hadronic form factors, commonly called G E (P2 ) and G M (P2 ), are needed for the description of the annihilation pro¯ and by varying the c.m. energy, their values cess e− e+ → ΛΛ, for time-like arguments s = P2 can in principle be determined. ¯ The joint Λ(→ pπ− )Λ(→ pπ ¯ + ) decay distributions were calculated and analyzed in Ref.[1], when the annihilation proceeds through an intermediate photon, using methods developed in [2, 3]. The interesting case of annihilation through an intermediate J/ψ or ψ(2S ) has been investigated in several theoretical [4, 5] and experimental papers [6, 7, 8]. This process has also been ¯ decay-asymmetry parameter used for determination of the Λ and for the CP symmetry tests in the hyperon system. A precise knowledge of the Λ decay parameter is needed for studies of spin polarization in Ω− and Ξ− decays. Presently, a collected data sample of 1.31 × 109 J/ψ events [9] by the BESIII detector [10] allows for high precision studies of spin correlations. In the experimental work referred to above, the jointhyperon-decay distributions employed were not the most general ones, but ones that were truncated. These truncated distribution functions do not permit a complete determination of the form factors and we therefore suggest to fit the experimental data to the general distribution described in [1]. Since the photon and the J/ψ are both vector particles, their corresponding annihilation processes will be similar. In fact, by a simple substitution, the cross-section distributions in Ref. [1], valid in the photon case, are transformed into distributions valid for the J/ψ case. An important aspect of these distributions is
Email addresses:
[email protected] (G¨oran F¨aldt),
[email protected] (Andrzej Kupsc) Preprint submitted to Elsevier
their generic nature, i.e. they are independent of the coordinate systems chosen for the hyperon decays. However, in order to compare with measured data, we need the distribution functions expressed in some specific coordinate system. For this purpose we employ the system introduced in [1]. However, for clarity, we also give its relation to other systems in use. 2. Basic necessities The annihilation process considered is described by the diagram of Fig.1, with momenta as indicated. The couplings of the initial-state leptons to the J/ψ are determined by the decay J/ψ → e+ e− . Assuming the J/ψ to decay via photons we can assume tensor couplings to be negligible. The vector coupling of the J/ψ to leptons is therefore the same as for the photon, if replacing the electric charge eem by a coupling strength eψ . From the decay J/ψ → e+ e− one derives αψ = e2ψ /4π =
3 Γ(J/ψ → e+ e− )/mψ . 2
e− (k1 )
(2.1)
Λ(p1 )
J/ψ(P )
e+ (k2 )
¯ 2) Λ(p
¯ when mediated by the J/ψ Figure 1: Graph describing the reaction e+ e− → ΛΛ meson.
In the current matrix element of the Lambda, both form factors are taken into account. We follow Ref. [1] and write the February 24, 2017
Lambda current as
T4
jµ (p1 , p2 ) =
=
√
1 − α2 cos(θ) cos(∆Φ).
(2.16)
−ie¯u(p1 )Oµ (p1 , p2 )v(p2 ), (2.2) 2M Oµ (p1 , p2 ) = G M (s)γµ − 2 G M s) − G E (s) Qµ , (2.3) Q
Only three of the five T functions are independent. The origin of the square root in Eqs.(2.11) and (2.16 ) is the relation
with P = p1 +p2 , and Q = p1 −p2 , and M the Lambda mass. The argument of the form factors is equal to s = P2 . In particular, at threshold where P2 = 4M 2 and Q2 = 0, we have G M = G E . A first attempt to calculate the hyperon form factors has been reported in [11]. The two form factors are often expressed in terms of two other variables, α and R,
(2.17)
G E (s) = R
=
Rei∆ΦG M (s), √ √ 1 − α s G E , = √ GM 1 + α 2M
√ 2M √ R(1 + α) = 1 − α2 , s easily obtained from Eq.(2.5). 3. Cross sections
The information we seek lies in the joint decay distributions ¯ as described by the diagram of Fig. 2. In the overall of Λ and Λ c.m. system, momenta are defined as,
(2.4) (2.5) (2.6)
where ∆Φ is relative phase between the G E (s) and G M (s) form factors. The α parameter is called asymmetry parameter of the ¯ decay, and is determined by experiments. J/Psi → ΛΛ Being a vector meson the J/ψ propagator takes the form gµν − Pµ Pν /m2ψ s−
m2ψ
+ imψ Γ(ψ)
,
=
α
=
s|G M |2 + 4M 2 |G E |2 , s|G M |2 − 4M 2 |G E |2 . s|G M |2 + 4M 2 |G E |2
ez
(3.20)
ˆ = p,
ey
=
ex
=
(3.21)
1 ˆ (pˆ × k), sin(θ) 1 ˆ × p. ˆ (pˆ × k) sin(θ)
(3.22) (3.23)
Expressed in terms of these basis vectors (2.8)
kˆ = sin(θ)e x + cos(θ)ez .
e− (k1 )
(2.12)
T1
= α + cos (θ),
(2.13)
T2
= −α sin (θ),
(2.14)
T3
= 1 + α,
(2.15)
Λ(l1 ) −
π (q1 ) J/ψ(P )
Λ(p1 ) ¯ 2) Λ(p
(2.10)
= 1 + α cos2 (θ),
(3.24)
This coordinate system is used for defining the directional angles of the decay proton in the Λ rest system. The spherical angles for the proton are denoted θ1 and φ1 .
(2.9)
T0
2
(3.19)
The scattering plane with the vectors p and k make up the xzplane, with the y-axis along the normal to the scattering plane. We choose a right-handed coordinate system with basis vectors
(2.7)
π + (q2 )
The structure functions depend on the scattering angle θ in ¯ The five functions the overall c.m. for the reaction e− e+ → ΛΛ. are, one denoted S and five denoted T , √ S = 1 − α2 sin(θ) cos(θ) sin(∆Φ), (2.11) 2
k1 = −k2 = k,
ˆ cos(θ) = pˆ · k.
In Ref. [1] polarizations and cross-section distributions were expressed in terms of five structure functions, themselves functions of the form factors G M and G E . Here, the form factors are replaced by the variable α of Eq. (2.5), which is used by the experimental groups [6, 7, 8] as well. We also extract a factor sD from all of the structure functions. The D and α functions are defined by D
(3.18)
and the scattering angle as
where Γ(ψ) is the full width of the J/ψ. However, by gauge invariance the contribution from the Pµ Pν term vanishes, leaving a gµν term which is the same in the photon case. As a result, the cross-section distribution for annihilation through J/ψ is obtained form the cross-section distribution for annihilation through the photon, by the replacement e2ψ e2em → . s2 (s − m2ψ )2 + m2ψ Γ2 (ψ)
p1 = −p2 = p,
e+ (k2 )
¯ 2) Λ(l
¯ Figure 2: Graph describing the reaction e+ e− → Λ(→ pπ− )Λ(→ pπ ¯ + ).
This same coordinate system is used for defining the direc¯ rest system. The tional angles of the decay anti-proton in the Λ spherical angles are then denoted θ2 and φ2 . The important G functions of Ref. [1] contain only rotation¯ decay products ally invariant scalar products. Since the Λ and Λ 2
¯ decay and for the Λ
are measured in the same coordinate system, those scalar products are easily calculated. We get G00
=
T0 ,
(3.25)
G05
=
αΛ S sin θ1 sin φ1 ,
(3.26)
50
=
(3.27)
G55
=
αΛ¯ S sin θ2 sin φ2 , αΛ αΛ¯ T1 cos θ1 cos θ2
G
(e2x , e2y , e2z ) = (e x , −ey , −ez ), We have transformed their spherical angles to our spherical angles, resulting in H 00 H 55
+T2 sin θ1 sin θ2 cos(φ1 − φ2 ) + sin θ T3 sin θ1 cos φ1 sin θ2 cos φ2 + sin θ T4 cos θ1 sin θ2 cos φ2 + sin θ1 cos φ1 cos θ2 . (3.28)
= =
U0 ,
(4.31)
αΛ αΛ¯ V1 cos θ1 cos θ2
2
+V2 sin θ1 sin θ2 cos(φ1 − φ2 ) , with structure functions
Remember that these G functions are a factor sD smaller than those of Ref. [1]. The normalization of the cross-section distribution is as follows, αem αψ ΓΛ ΓΛ¯ 1 p · dΓ = 2 2 2 2 2 64π k (s − mψ ) + mψ Γ (ψ) Γ2 (M) X ab · D G dΩΛ dΩ1 dΩ2 . (3.29)
U0
=
1 + α cos2 (θ),
(4.33)
V1
=
α + cos (θ),
(4.34)
V2
=
− 12 (1
2
− α) sin (θ). 2
(4.35)
We note the absence of terms corresponding to S, T3 , and T4 . Hence, in our view, a truncated distribution. Also, the coefficient for T2 is different. 5. Comparison with the truncated distribution
a,b
Here, k and p are the initial- and final-state momenta in the ¯ The decay angles Ω1 and Ω2 are angles reaction e+ e− → ΛΛ. ¯ and ΩΛ in the global measured in the rest systems of Λ and Λ, c.m system. ΓΛ and ΓΛ¯ are the channel widths and Γ(M) and Γ(ψ) total widths. If we integrate over the lambda-decay angles Ω1 then the contributions from the functions G05 and G55 are annulled [2], and correspondingly for the angles Ω2 . There is another important remark to be made. The S and T functions depend only on α and ∆Φ. This means they only determine the ratio |G E /G M | of the form factors, and not their absolute values. For this it is necessary to determine the D function of Eq.(2.9), which according to Eq.(3.29) can be obtained from a measurement of the magnitude of the cross section distribution. For this purpose one integrates the cross-section distribution over the decay angles Ω1 and Ω2 , yielding the unpo¯ larized cross-section distribution for the reaction e+ e− → ΛΛ.
We shall now illustrate how the application of the above expressions affects the values of the extracted parameters. For this purpose, we compare the cross-section distributions of Eq.(3.29) and Eq.(4.32), taking the latter from Ref.[8]. Assuming zero relative phase, ∆Φ = 0, we have ( ) 1 d(Γ − ΓB ) 3αΛ αΛ¯ sinθ = Γ d cosθ dΩ1 dΩ2 64π2 (3 + α) × sinθ sinθ1 sinθ2 2(1 − α) cos(φ1 − φ2 ) + (1 + α) cos(φ1 + φ2 ) √ +2 1 − α2 cosθ sinθ1 cosθ2 cosφ1 + cosθ1 sinθ2 cosφ2 (5.36) Fig. 3 displays the relative difference (|M|2 − |MB |2 )/|M|2 for a uniform scan of phase space assuming CP conservation, αΛ¯ = −αΛ , and using αΛ = 0.642 [12] and α = 0.469 [9]. The importance of the extra terms in Eq.(3.29) for the analysis of the experimental data could be also illustrated by evaluating the following moments: Z 1 dΓ Mi ≡ Fi (θ, Ω1 , Ω2 ) d cos θ dΩ1 dΩ2 , Γ d cosθ dΩ1 dΩ2 (5.37) where the functions Fi (θ, Ω1 , Ω2 ) are defined as
4. Simplifed distribution Until today, the cross-section distributions described above have not been confronted with experimental data [6, 7, 8]. For this purpose a simplified distribution was in all cases employed. Its application and structure is described in [8], and its nonnormalized cross-section distribution is dΓB = N0 (H 00 + H 55 ) dΩΛ dΩ1 dΩ2 .
(4.32)
F1 (θ, Ω1 , Ω2 ) = sin θ cos θ (5.38) (sin θ1 cos θ2 cos φ1 + cos θ1 sin θ2 cos φ2 ) ,
(4.30)
A derivation using helicity amplitudes can, e.g., be found in [5]. It is quite different from ours. In Ref. [8] the coordinate systems used for the hyperon decays differ from ours, Eqs. (3.21), (3.22), and (3.23). The relation between the basis vectors are, for the Λ decay
F2 (θ, Ω1 , Ω2 ) = sin2 θ sin θ1 sin θ2 cos(φ1 + φ2 ).
(5.39)
The moments become √ 4 1 − α2 M1 = αΛ αΛ¯ , 45 α + 3
(e1x , e1y , e1z ) = (−e x , −ey , ez ), 3
(5.40)
a.u.
400
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×103
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
350 300 250 200 150 100 50 0 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
(|M|2-|MB|2)/|M|2
Figure 3: Relative difference between Eq.(3.29) and Eq.(4.30) of Ref. [8]. The squared ampitude is obtained from a uniform scan of phase space.
M2 =
8 α+1 αΛ αΛ¯ . 45 α + 3
(5.41)
Both moments vanish for the cross-section distribution of Ref. [8]. One can express the expected errors of these moments as a function of number of collected events N. Here we quote the values using the same assumptions as for creating Fig. 3, M1 = −0.00933, M2 = −0.031,
0.175 ∆M1 = √ , N 0.33 ∆M2 = √ . N
This means that for an experiment with 10000 events and flat acceptance one expects 5σ and 9σ deviation from zero for M1 and M2 respectively. The additional terms when ∆Φ , 0 are (we keep αΛ¯ = −αΛ ), √ ( ) 1 d(Γ − Γ∆Φ=0 ) 3αΛ 1 − α sinθ cosθ = Γ d cosθ dΩ1 dΩ2 64π2 (3 + α) ∆Φ × 2αΛ sin2 sin θ1 cos θ2 cos φ1 + cos θ1 sin θ2 cos φ2 2 + sin(∆Φ) sin θ1 sin φ1 − sin θ2 sin φ2 . (5.42) ∆Φ can be extracted from the expectation value of the function F3 (θ, Ω1 , Ω2 ) = cos θ sin θ(sin θ1 sin φ1 − sin θ2 sin φ2 ). (5.43) The corresponding moment is √ 4 1 − α2 M3 = − αΛ sin(∆Φ). 15 α + 3
(5.44)
Using the numerical values of α and αΛ mentioned above, one √ can estimate the sin(∆Φ) uncertainty as 6.8/ N. Acknowledgments We are grateful to Tord Johansson who provided the inspiration for this work. 4
