Formalismus in helicity and invariant amplitudes

With the help of the T-matrix of Eq. (2.13) represented by the helicity amplitudes or of the relation between the amplitudes Hi and Ai of Eq. (D.1), one can get the observables in terms of the helicityH_i or the invariantA_i amplitudes.

4. Polarized nucleon Compton scattering

The c.m. unpolarized differential cross section d¯σ/dΩ^∗ reads then d¯σ As shown in Eq. (4.38) an unpolarized differential cross section is simply given as the sum of a squares of the individual helicity amplitude |Hi|², and is then averaged over the four possible initial states. The |H_i|² is not measurable directly because the polarizations in a final state are determined by the interaction between the particles.

Thus, it is allowed to measure only the different linear combinations of them, i.e dif-ferential cross sections or asymmetries. The explicite expressions of the single and double polarization observables are as follows:

dσ¯

4. Polarized nucleon Compton scattering

4. Polarized nucleon Compton scattering

As said, H₃ = H₄ = H₅ = H₆ = 0 at θ = 0 (t = 0) and H₁ = H₂ = H₃ = H₄ = 0 at θ = π(η = 0). In Eqs. (4.38)-(4.53), it is hence simply recognizable that all observables, excepting the c.m. unpolarized differential cross section, Σ1z and Σ2z, are equal to zero at these two extreme angles. Notice also that below the pion threshold (ω= 150 MeV) there are only three asymmetries, Σ₃, Σ_2x, and Σ_2z, which are different from zero. Indeed, Σ2z is the only measurable quantity in this region of energy and at the anglesθ = 0 andπ. From

On the other hand, the cross section d¯σ/dΩ^∗ and the asymmetry Σ_2z are also deter-mined by the amplitudes A_3,4 and A₆ at the angle θ = 0 and by the amplitudes A_1,2

4. Polarized nucleon Compton scattering In order to investigate the sensitivity of the observables to the invariant amplitudes, we turn the strength of the each invariant amplitude A_i from 100% to 101% and illustrate the difference between the results by using the invariant amplitude with the unchanged strength and that obtained from the amplitudes A_i having increased strength in Fig. 4.1-4.6 by means of the solid (θ = 30⁰), dashed (θ = 60⁰), dot-ted (θ = 120⁰) and dashed-dotted lines (θ= 150⁰), respectively.

At the small angles (θ= 30⁰ and 60⁰) the maximum respone to a change of A_3,4 and A₆ is, as expected, shown in the asymmetry Σ_2z. Furthermore, the effects of the am-plitudesA₃ and A₆ on Σ_2z are most visible just above and below the pion threshold, i.e. ω '130 MeV as well as about ω ' 170 MeV. Since the sum of the A₃ and A₆ is connected with theα+β at the angleθ = 0, the measurment of the asymmetry Σ_2z in this kinematical region could offer important information on the electromagnetic po-larizability. On the other hand, the influence of changing theA₄ amplitude is seen not only for Σ_2z at higher energies, but also for Σ_1x at the large angleθ = 120⁰ and above the energy of the second resonance (ω '750 MeV). The sensitivity of the amplitudes A₃ and A₆ are generally observed in the low energy region and at the small angles in nearly all asymmetries. In the case of the unpolarized differential cross section, the dependence on the amplitudes A_3,4 and A₆ is specifically large at ω ' 320 MeV and around the energies ω '750 MeV, respectively.

At the large backward angles of θ = 120⁰ and 150⁰ the change of the strength of the amplitudes A₂ and A₅ has the largest influence on Σ_2z at the energy ω ' 730 MeV.

This means that the measurment of the asymmetry Σ_2z in this high energy region and at the larger angles is appropriate to get more knowledge on the proton backward spin polarizability γ_π^p. In the case of the amplitude A₁, the asymmetry Σ_2z as well as Σ₃ are sensitive to a change of the strength of this amplitude, in particular below the one-pion threshold and in the region of ∆-resonance, respectively. Among the three amplitudes A₁, A₂ and A₅, the amplitude A₁ affects mostly the unpolarizaed differential cross sectionsd¯σ/dΩ^∗.

As a result, the experiment using the circularly polarized photon and the proton-target polarized inz direction is mostly adequate to determine the value of the polarizabili-ties.

4. Polarized nucleon Compton scattering

Deviations ^{Θcm =30}

o

Θ_cm =60^o Θ_cm =120^o Θ_cm =150^o dσ/dΩ

Σ_y

Σ₃ Σ_3y

Σ_2x

Σ_1x

E_γ(MeV)

Σ_2z

E_γ(MeV)

Σ_1z -100

-50 0 50

-0.1 -0.05 0 0.05 0.1

-0.2 0 0.2

-0.2 0

-0.2 0

-0.1 0 0.1 0.2 0.3

-0.5 0

0 150 300 450 600 750 900 -0.2 -0.1 0

150 300 450 600 750 900

Figure 4.1.:The relative difference between the results for the observables of proton Compton scattering obtained from the invariant amplitude A₃ with the unchanged strength and that obtained from the amplitudes A₃ with the strength increased by 1% at four different angles θ_cm = 30^o (solid lines), 60^o (dashed lines), 120^o (dotted lines) and 150^o (dashed-dotted lines).

4. Polarized nucleon Compton scattering

Deviations ^{Θcm =30o}_Θ

cm =60^o Θ_cm =120^o Θ_cm =150^o dσ/dΩ

Σ_y

Σ₃

Σ_3y

Σ_2x Σ_1x

E_γ(MeV)

Σ_2z

E_γ(MeV)

Σ_1z -15

-10 -5 0

0 0.02

-0.02 -0.01 0 0.01

-0.04 -0.02 0 0.02 0.04

-0.04 -0.02 0 0.02 0.04

-0.025 0 0.025 0.05

-0.1 -0.05 0 0.05

0 150 300 450 600 750 900

-0.01 0 0.01 0.02

150 300 450 600 750 900

Figure 4.2.:The relative difference between the results for the observables of pro-ton Comppro-ton scattering obtained from the invariant amplitude A₄ with the unchanged strength and that obtained from the amplitudes A₄ with the strength increased by 1% at four different angles θ_cm = 30^o, 60^o 120^o and 150^o. The notations are the same as in Fig. 4.1.

4. Polarized nucleon Compton scattering

Deviations

Θ_cm =150^o Θ_cm =120^o Θ_cm =60^o Θ_cm =30^o dσ/dΩ

Σ_y

Σ₃

Σ_3y

Σ_2x Σ_1x

E_γ(MeV)

Σ_2z

E_γ(MeV)

Σ_1z -100

-50 0

-0.05 0 0.05

-0.2 0 0.2

-0.1 0 0.1

-0.1 0 0.1 0.2

-0.2 -0.1 0 0.1

-0.5 0 0.5

0 150 300 450 600 750 900

-0.1 0 0.1

150 300 450 600 750 900

Figure 4.3.:The relative difference between the results for the observables of pro-ton Comppro-ton scattering obtained from the invariant amplitude A₆ with the unchanged strength and that obtained from the amplitudes A₆ with the strength increased by 1% at four different angles θ_cm = 30^o, 60^o 120^o and 150^o. The notations are the same as in Fig. 4.1.

4. Polarized nucleon Compton scattering

Deviations

dσ/dΩ

Θ_cm =30^o Θ_cm =60^o Θ_cm =120^o Θ_cm =150^o

Σ_y

Σ₃ Σ_3y

Σ_2x Σ_1x

E_γ(MeV)

Σ_2z

E_γ(MeV)

Σ_1z -40

-30 -20 -10 0

-0.02 0 0.02

-0.05 -0.025 0 0.025 0.05

-0.02 0 0.02

-0.04 -0.02 0 0.02

-0.04 -0.02 0 0.02

-0.05 0 0.05

0 150 300 450 600 750 900

-0.01 0 0.01 0.02

150 300 450 600 750 900

Figure 4.4.:The relative difference between the results for the observables of pro-ton Comppro-ton scattering obtained from the invariant amplitude A₁ with the unchanged strength and that obtained from the amplitudes A₁ with the strength increased by 1% at four different angles θ_cm = 30^o, 60^o 120^o and 150^o. The notations are the same as in Fig. 4.1.

4. Polarized nucleon Compton scattering

150 300 450 600 750 900

Figure 4.5.:The relative difference between the results for the observables of pro-ton Comppro-ton scattering obtained from the invariant amplitude A₂ with the unchanged strength and that obtained from the amplitudes A₂ with the strength increased by 1% at four different angles θ_cm = 30^o, 60^o 120^o and 150^o. The notations are the same as in Fig. 4.1.

4. Polarized nucleon Compton scattering

150 300 450 600 750 900

Figure 4.6.:The relative difference between the results for the observables of pro-ton Comppro-ton scattering obtained from the invariant amplitude A₅ with the unchanged strength and that obtained from the amplitudes A₅ with the strength increased by 1% at four different angles θ_cm = 30^o, 60^o 120^o and 150^o. The notations are the same as in Fig. 4.1.

4. Polarized nucleon Compton scattering

Im Dokument Polarized Compton Scattering off the Nucleon (Seite 37-48)