Outlook: Systematic approach to the rescattering

In this section we present a construction of the theoretical framework based on unitarity and analytic continuation which will allow to systematically address the final-state interaction (rescattering) in a system of three particles. The approach was named after Khuri and Treiman who first applied it for the decayK →3π[148]. The formalism was further developed in Ref. [53, 149, 162–164]. The idea of the approach is to enforce thetwo-bodyunitarity relation to the subchannel interactions. Starting with the general isobar ansatz (3.7) we are going to relate the discontinuity of the isobar partial wave amplitude to the cross channel projections. A construction based on the dispersive relations leads to integral equations for the isobar amplitude. The total invariant mass is used as a tuning parameter which connects the physical scattering domain whensis small with the decay domain whensgoes above the three-particle threshold.

The KT technique has been already applied to many decays: ω/φ[165, 166],η→3π[167–169], η⁰ →ηππ[170],D⁺→K⁻π⁺π⁺[171],D⁺→K_Sπ⁰π⁺[172]. However, it remains a state of art since for every individual case the formalism differs a little bit due to a different totalJ^{P C} of the system and different sets of isobar partial waves contributing to the final state. Here, we demonstrate how the formalism can be generalized using the advantages of helicity basis andLS-decomposition.

We calculate the cross channel projections for arbitrary partial waves in a system of three pions under exact isospin symmetry.

A common procedure to build the model for the decay amplitude starts by writing all possible covariant structures which contract the polarization tensor of the decaying particle momenta of the final-state particles. Every covariant construction is supplied with a scalar amplitude which is a subject for the unitarity constraints. For example [9, 165, 173]:

A_ω→3π =A(σ₁, σ₂, σ₃)_αβγδ^α_ωp^β₁p^γ₂p^δ₃, (4.17) A_a1→3π =B(σ₁, σ₂, σ₃)^µ_a1(p₁−p₂)_µ+C(σ₁, σ₂, σ₃)^µ_a1(p₂−p₃)_µ, (4.18) where A, B, and C are scalar functions. The main reason of complications in the customary construction is that one needs to calculate the partial wave projections for every tensor structure which appears in the covariant amplitude.

As a rather general case, we consider a three-pion production amplitude, where possible values of the total angular momentumJ of the3πsystem, with a projectionM, are allowed. (The diffractive

4.5 Outlook: Systematic approach to the rescattering

productionπ p→3π p⁰, measured at COMPASS is a good example of the reaction where all values ofJ andMare allowed) We use the same notations for the three-particle partial waves as discussed in Sec. 3.3.1, and in Appendix C.1: Sis the orbital angular momentum ofππ,Lstands for the relative orbital momentum between theππsubsystem and the remaining pion. We will, however, drop the isospin coefficients for simplicity. These coefficients are presented in Appendix C.1, and it is not difficult to bring them back. Theproductionamplitude is denoted by A(τ)(cf. Eq. (3.5)), where the kinematic variablesτ are the subchannel invariant mass squared,σ_iand two pairs of spherical angles as discussed in Eq. (3.2). We will omit thetandsdependence of the amplitudeAas these quantities enter as parameters in the formalism. The general constraint on thesdependence is subject of Chapter 7.

An expansion of the three-particle state leads to the general isobar decomposition (the decomposition for theπ⁻π⁺π⁻system was discussed in Sec. 3.3.1, see Eq. (6.6) with Eq. (3.7) and Eq. (3.7)):

A(τ) = X

JM LS

F_LS^JM(σ₁)Z_LS^JM∗(Ω₁,Ω₂₃)

+F_LS^JM(σ₂)Z_LS^JM∗(Ω₂,Ω₃₁) +F_LS^JM(σ₃)Z_LS^JM∗(Ω₃,Ω₁₂)

, (4.19) where the angular functionZ_LS^JM is given by Eq. (3.9). Every isobar partial wave series is truncated for all channels(1),(2), and(3). We notice here that the full amplitude includes an infinite number of partial waves whatever channel is considered due to the cross-channel projections of the isobar partial wave. In Eq. (4.19), the same partial wave projected isobar amplitudeF_LS^JM enters to all channels since the final-state particles are identical.

The projection of the full amplitude to the channel(1)partial waves is found by integrating over two pairs of the spherical angles,Ω₁andΩ₂₃:

A^JM_LS (σ₁) =

Z dΩ1

4π dΩ23

4π Z_LS^JMA(τ) (4.20)

=F_LS^JM(σ₁) + Z dΩ₁

4π dΩ₂₃

4π Z_LS^JM X

J⁰M⁰L⁰S⁰

h

F_L^J0⁰S^M00(σ₂)Z_L^J0⁰S^M00∗(Ω₂,Ω₃₁) + (−→123)i

| {z }

Fˆ_LS^JM(σ₁)

,

where Fˆ_LS^JM is a projection of cross channels, (−→123)indicates a symmetric term obtained by the 1→2→3→1permutation from the first term in the square bracket. The integral in Eq. (4.20) can be simplified using properties of the Wigner D-functions and relations between rotations which are discussed in Ref. [91, 96] and are detailed in Appendix C.3. It reads:

X

λ

D^J_{M λ}(φ₃, θ₃,0)D_λ0^S (φ₁₂, θ₁₂,0) =X

λ

D_{M λ}^J (φ₃, θ₃, φ₁₂)d^S_λ0(θ₁₂)

=X

λν

D_{M ν}^J (φ₁, θ₁, φ₂₃)d^J_νλ(ˆθ₃)d^S_λ0(θ₁₂)

=X

λν

D_{M λ}^J (φ₁, θ₁, φ₂₃)d^J_λν(ˆθ₃)d^S_ν0(θ₁₂). (4.21)

The integrals over dθ₁, dφ₁, and dφ₂₃in Eq. (4.20) drop due to orthogonality of theD-functions.⁵ We can also immediately conclude that only waves with the sameJ⁰ =J,M⁰ =M contribute to the inhomogeneous term ofFˆ_LS^JM.

Fˆ_LS^JM(σ₁) =X

L⁰S⁰

X

λν

s(2L+ 1)(2S+ 1)(2L⁰+ 1)(2S⁰+ 1)

(2J + 1)² hL,0;S, λ|J, λi

L⁰,0;S⁰, ν |J, νi

×

Z dcosθ₂₃ 2

F_L^JM0S⁰(σ₃)d^S_λ0(θ₂₃)d^J_λν(ˆθ₃)d^S_ν0⁰(θ₁₂)

+ (−1)^λ+νF_L^JM0S⁰(σ₂)d^S_λ0(θ₂₃)d^J_λν(ˆθ₂)d^S_ν0⁰(θ₃₁)

, (4.23) where the anglesθˆ₃,θ₁₂,θˆ₂, andθ₃₁can be expressed as functions ofσ₁andθ₂₃, or equivalently as functions of invariantsσ₁,σ₂andσ₃as shown in Appendix C.3.

The unitarity constraint for the amplitudeA^JM_LS analytically continued from the scattering domain reads:

d_σA^JM_LS (s, σ) =t^†_S(σ)ρ(σ)A^JM_LS (s, σ), (4.24) wheret_S(σ) is the partial wave projected ππ scattering amplitude (to avoid confusion with the (S= 0)-wave we remind the reader thatSstands for the(ππ)-orbital angular momentum). Eq. (4.24)

is nothing else but the production unitarity equation discussed in Sec. 2.3,cf. Eq. (2.34).

To relate the discontinuity of the amplitudeA^JM_LS tod_σF_LS^JM we need to make one assumption on the analytic structure ofF_LS^JM: the inhomogeneous termFˆ_LS^JM does not contribute to the right-hand-side discontinuity ofA^JM_LS . The unitarity equation for the isobar partial wave becomes, (cf. Eq. (2.37))

d_σF_LS^JM(σ) =i t^†_S(σ)ρ(σ) (F_LS^JM(σ) + ˆF_LS^JM(σ)). (4.25) This equation can be inverted the same way as we demonstrated in Sec. 2.3. As a demonstration, we make a few unnecessary assumptions and write the final form of the integral equation.

1. t_S(σ)does not have a left-hand cut.

2. Fˆ_LS^JM(σ)vanishes faster than1/σwhenσ → ∞.

3. For the demonstration purposes, we are going to ignore the kinematic singularities of the of the functionF_LS^JM(σ)andt_S(σ). ⁶

5As a remark we notice that by applying a transformation from Eq. (4.21) to the original decomposition in Eq. (4.19), a common rotation functionD^J_{M λ}^∗ (φ₁, θ₁, φ₂₃)can be pulled out of the square bracket:

A(τ) = X

JM LS

p(2L+ 1)(2S+ 1)X

λν

D_{M λ}^J∗(φ₁, θ₁, φ₂₃)×

F_LS^JM(σ₁)hL,0;S, λ|J, λiδ_λνd^S_ν0(θ₂₃)

+F_LS^JM(σ₂)hL,0;S, ν |J, νid^S_λν(ˆθ₃)d^S_ν0(θ₁₂)

+F_LS^JM(σ₃)(−1)^λ+νhL,0;S, ν|J, νid^S_λν(ˆθ₂)d^S_ν0(θ₃₁)

. (4.22)

When the orientation of the3πproduction plane is not measured, the common rotation factor can be omitted.

6A consistent treatment of these singularities is, perhaps, the most tedious part of the problem (e.g.see the analysis of the kinematic singularities inBdecays [2] andΛ_bdecays [13])

4.5 Outlook: Systematic approach to the rescattering

Figure 4.17: Diagrammatic representation of theKTequations (see Eq. (4.26)). The thick red line represents the amplitudeFLS^JM, the green double line denotestS(σ), the large black circle replacesCLS^JM.

The equation for the amplitudeF_LS^JM then reads:

F_LS^JM(σ) =t_S(σ)

"

C_LS^JM(σ) + 1 2π

Z _∞

4m²_π

ρ(σ⁰) ˆF_LS^JM(σ⁰) σ⁰−σ dσ⁰

#

, (4.26)

whereC_LS^JM(σ)is an entire function ofσ. The inhomogeneous term,Fˆ_LS^JM, which stands under the integral, is related toF_LS^JM itself by Eq. (4.23) (see also Fig. 4.17), therefore, Eq. (4.26) is an integral equation. We see that these equations have a recursive form, and hence, can be solved by iterations.

Let us suppose thatC_LS^JM = 1, then, on the first interaction(F_LS^JM)⁽⁰⁾takes the form of the known(ππ) amplitude,t_S(σ), for all subchannels. The partial wave projection to one of the channels, however, includes not only the direct channelt_S(σ), but also cross channel projections, Eq. (4.20). Such a partial wave projection amplitude cannot satisfy the unitary sincet_S(σ)by itself does. Hence, we have to modify the amplitude(F_LS^JM)⁽⁰⁾in order to compensate (see discussion about Schmid cancellation in Sec. 4.2.4) for the cross channel projections, by adding a triangle diagram (second iteration). Since the cross channel amplitudes are modified accordingly, the projections of them change and we have to add the compensation term again which leads to the third iteration, and so on.

By a simple comparison, one can validate that the expression for the triangle diagram, calculated using the dispersive approach in Sec. 4.2.3, matches the first iteration of Eq. (4.26). The inhomogeneous term is a partial wave projection to theKK-channel of the cross channel, with¯ Kπamplitude in the πKK¯ system.

C H A P T E R 5

Im Dokument Three-pion dynamics at COMPASS: (Seite 80-85)