Decomposition in the COMPASS basis

contributions from the different waves:

I_3π^B(s, t)≈√

s X

JM λS

Z B^JM_λS Z_λS^JM2dΦ₃ (5.25)

=X

J

√s (8π)²s

Z ₍^√_s−mπ)²

4m²_π

dσ 2π

λ^1/2(σ, s, m²_π)λ^1/2(σ, m²_π, m²_π) σ

X

M λS

B^JM_λS (t, s, σ)², (5.26) where we neglected interference terms to simplify the phase space integration. This approximation affects theJ^{P C} = 0⁻⁺sector the most, its impact to the other sectors is small (one sees it indirectly by comparing the left and the middle panels of Fig. 5.6).

We normalize the models (a single normalization constant for every of the three models) to to contribute a certain fraction of the total intensity of the COMPASS data. This normalization factors scales all projections simultaneously. This fraction is set to 30% as a compromise between an adjustment of theJ^{P C} = 4⁻⁺andJ^{P C} = 3⁻⁺as shown in Fig. 5.5.

All projections have a bump-like structure: there is a low-energy rise which is related to the threshold behavior; the intensity distribution peaks at the region1. . .2 GeV, as higher as biggerJ is;

then, it starts falling and approaching its asymptotic limit. The “Standard Deck” model produces the broadest structures for all projections. For this model, the sum of waves withJ = 0is huge above 2 GeV, that clearly contradicts the data. A significance of theJ = 0would mean that intensity does not vanish for thecosθ₁= 0(as in Fig. 5.4), while it is the opposite in the data (seecosθ_GJ×m_3π distribution in Fig. 3.8).

The models II and III have similar projections which suggest a reasonable background for J^{P C} = 1⁺⁺,2⁻⁺. They also give a good justification for theJ^{P C} = 4⁺⁺and3⁻⁺distributions.

For theJ^{P C} = 0⁻⁺, the enhancement in the data at1.3 GeVcould be suspected to be theπ(1300), previously observed inpp¯ande⁺e⁻annihilation as well as in diffraction [26]. ² However, we see that a large fraction of this peak have to be attributed to the background. An interesting question for the further investigation is a sensitivity of this intensity to the parameters of the model. A non-negligible intensity of the Deck process is found in the exotic sector J^{P C} = 1⁻⁺. It is in agreement with the result of the mass-dependent fit in Ref. [3] where a phenomenological parametrization of the background was used. The background intensity at 1.2 GeV hinders precise extraction of the resonance parameters of the exoticπ₁(1600).

5.4 Decomposition in the COMPASS basis

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

5.0×10⁴ 1.0×10⁵ 1.5×10⁵ 2.0×10⁵ 2.5×10⁵

1⁺⁺

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

1×10⁴ 2×10⁴ 3×10⁴ 4×10⁴

5×10⁴ 0⁻⁺

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

2000 4000 6000

8000 1⁻⁺

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

1×10⁴ 2×10⁴ 3×10⁴ 4×10⁴ 5×10⁴

2⁺⁺

Standard(I) Reggeized(II) Form−Factor(III)

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

2×10⁴ 4×10⁴ 6×10⁴

8×10⁴ 2⁻⁺

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

100 200 300 400

3⁻⁺

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

2500 5000 7500 10000

3⁺⁺

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

1000 2000 3000

4000 4⁻⁺

0.5 1.0 1.5 2.0 2.5

√s≡m3π(GeV) 0

1000 2000

3000 4⁺⁺

Figure 5.5: A comparison of intensities for the mainJ^{P C}from the COMPASSPWAand the calculations in three models for the Deck process. Partial wave intensities from the COMPASSPWA[78] summed for different J^{P C}sectors are overlapped by colored lines that correspond to the three Deck models discussed in the text.

Data points are presented by the black histograms with gray errors. The Deck models are normalized to have 30%of total intensity of the data.

functionsΨ_wdefined in Eq. (3.15). The approximation reads:

B(t, s, τ)'X

w

B_w(t, s)Ψ_w(s, τ). (5.27) The sign'implies that we need to find the coefficientsB_w(t, s)which approximate the original amplitudeBthe best. It is important to agree how the measure of the mismatch is defined. In the PWAtechnique, theintensityis approximated: my minimizing likelihood, effectively the distance between the absolute value between the left and the right part of Eq. (5.27) is minimized. This numerical procedure is rather involved. ³ For the studies in this section we use another strategy, algebraic-projectionmethod, in which we approximate theamplitudedirectly from Eq. (5.27). A system or linear equations for the coefficients B_w can be obtained by multiplying both parts of Eq. (5.27) withΨ^∗_qand integrating over the phase space. The coefficients are found by inverting the integral matrix:

B_w(t, s) =X

q

B_wq⁻¹(t, s) Z

B(t, s, τ)Ψ^∗_q(s, τ)dΦ3, (5.28) where dΦ₃is a three-body phase space integral,B_wqin the integral matrix defined in Eq. (3.20). The main difference of this algebraic method to the analytic method discussed above is the treatment of the subchannel energies. As for thePWAtechnique, we take the basis where the subchannel energy dependence is fixed (byf_S(σ)in Eq. (3.15)). There are still two methods to proceed which deal differently with the symmetrization (see Eq. (5.3)) of the amplitude:

1. project the complete, symmetrized amplitude from Eq. (5.3) to the symmetrized basis from Eq. (3.15);

2. project only one termB⁽¹⁾to the part of the basis which contains the partial-wave expansion in the corresponding subchannel. In this method, Eq. (5.28) have to be accordingly modified;

the integral matrix becomes orthogonal for waves with distinguishableJ^{P C}M, LS quantum numbers.

We call the first method, thesymmetrizedprojections, while the second is callednon-symmetrized. One may wonder what happens if theσ_kdependence of the amplitudeB^(k)is different to the one in our basis. For the non-symmetrized method, we obtain exactly the same projections since four angular integrals in dΦ₃ already guarantee orthogonality. However, the intensity are different since the specific line shape of the subchannel resonances, modified by the pion exchange, is not captured by the basis. In case of the symmetrized projections, the parasitic effects related to the fixed isobars are magnified due to the absence of strict orthogonality.

We restrict the discussion to the “Standard Deck” (Model-I) and address effects which appear due to the specific truncated basis. We can disentangle three issues:

1. A fixed shapes of the isobars in the basis lead to problems in description of the Deck model.

Significant artifacts appear for waves where theππsubsystem is in theS-wave.

3The partial-wave fit of the Deck process was pioneered by Ascoli et al. [83]. It was explored for the COMPASS kinematics by F. Haas and D. Ryabchikov Ref. [3, 77, 184]. The most of effects discussed in this section are also present in results of the fit method.

5.4 Decomposition in the COMPASS basis

2. Symmetrization of the amplitude and the basis leads to a correlation between waves with the variousLandSnumbers within the sameJ^{P C}.

3. “Threshold effects”: the number of waves in thePWAmodel changes from bin-to-bin, it results in the discontinuities of the waves’ intensities.

To separate those problems we consider three procedures. The simplest and cleanest procedure is the algebraic projection of the non-symmetrized amplitude to the non-symmetrized basis. Next, we turn on the thresholds and check their influence. Lastly, we perform an expansion of the symmetrized model by the symmetrized basis.

We use theMCmethod to calculate the integral on the right side of Eq. (5.28). The amplitude B_w(t, s)is discretized in100bins for√

s≡m_3πin the interval from0.5 GeVto2.5 GeV, the value of the transferred momentum is fixed,t=−0.1 GeV². We generate10⁶events distributed according to the phase space in eachm_3π bin. For every event we calculate the value of the Deck amplitude B(t_e, s_e, τ_e)as well as complex values for all basis functionsΨ_w(s_e, τ_e)where the subindexepoints to a specific evente. The integral is calculated analogously to Eq. (3.20a). The coefficientsB_ware found for every bin independently.

The partial wave intensities as a function ofm_3π are shown in Fig. 5.6 in the same fashion as Fig. 3.13. As mentioned already, the intensities in the non-symmetrized projection method on the

0.5 1.0 1.5 2.0 2.5

√s≡m_3π(GeV) 0.0

0.5 1.0 1.5

Intenisity,d2I/dtd√ s(a.u.) Symmetrized

sum 0⁻⁺ 1⁺⁺

2⁻⁺ 3⁺⁺

1⁻⁺

2⁺⁺

0.5 1.0 1.5 2.0 2.5

√s≡m_3π(GeV)

Non-symmetrized

0.5 1.0 1.5 2.0 2.5

√s≡m_3π(GeV)

Non-symm. with thresholds

Figure 5.6: The intensities of theJ^{P C}-sectors summed over the partial waves in the COMPASS basis. The left plot shows projections of the non-symmetrized Deck model to the non-symmetrized basis, the middle plot shows the projections of the symmetrized Deck model to the symmetrized basis. The right plot shows the effect of the “thresholding” while the amplitude as well as the basis is kept non-symmetrized.

middle panel of Fig. 5.6 agree with the analytic projections due to the orthogonality of the partial waves with respect to the all quantum numbers (JM LS). However, the basis contains several waves withS = 0which share exactly the same quantum numbers but differ by the line shape of the ππ-amplitude. The line shape of the[ππ]_S Deck amplitude (see Fig. 5.2) is similar to theππ→ππ amplitude shown in the left panel of Fig. 3.16 (the difference comes from theσ-dependence of the pion propagator), while in the basis we have three functions with the predetermined line shapes: (ππ)_S, f₀(980), andf₀(1500). The decomposition obtained in the algebraic-projection method is shown in Fig. 5.7. The description is far from being perfect: the shape is tilted and contains artifacts. As we can see on the left panel of the Fig. 5.7, thef₀(1500)Breit-Wigneramplitude is eventually preferred over

0.2 0.4 0.6 0.8 1.0

√σ≡m_π+π⁻(GeV) 0

25 50 75 100 125

Intenisity,d3I/dtdsd√ σ(a.u.)

π⁺π⁻ spectrum for m_3π = 1.21 GeV

Sum ρ f2

f₀(1500) (ππ)S

f0(980) ρ₃ Total

0.3 0.6 0.9 1.2 1.5

√σ≡m_π+π⁻(GeV) 0

10 20 30

π⁺π⁻ spectrum for m_3π = 1.91 GeV

Figure 5.7: Description of theππ-spectrum of the Deck amplitude by the partial-wave model in the COMPASS basis that is found by the algebraic-projection method (the non-symmetrized Deck to the non-symmetrized basis). The intensity of the coherent sums of the wavesJ^{P C}Mξπ Lwith the sameIsobarξare shown by different colors. To plot the total intensity distributions we use theMCphase-space sample and weightsw_e calculated for every event. For the total intensity shown by the red linew_e=|B(t_e, s_e, τ_e)|², while for the other distributionsw_e=|PB_w(t_e, s_e, τ_e)Ψ_w(s_e, τ_e)|².

the(ππ)_S-Isobarfor the description of theπ⁺π⁻spectrum of the Deck. It also causes a prominent f₀(1500)peak in theππspectrum which is not present in the original Deck model on the right panel of the Fig. 5.7.

Problems in description of the amplitude lead to deviations of the integral intensity from the one calculated in the model (in contrast to thePWAtechnique, the number over events per bin is not constrained). The largest effect is observed in theJ^{P C} = 0⁻⁺sector as can be seen by comparing the green distribution in the middle panel of Fig. 5.5 with the orange line in the middle-top panel of Fig. 5.7. The splitting of the total0⁻⁺ intensity into contributions of the individual waves is presented in Fig. 5.8. One finds a peak around 1 GeV on the right panel in bothf₀(980)π S-wave andf₀(1500)π S-wave, however, not in the total intensity. These two waves largely interfere in the expansion series. It has nothing to do with the physics: due to the pure model for the scalar waves, the found decomposition is preferred in the numerical procedure. The left plot of Fig. 5.8 shows how the decomposition changes when the wave withf₀(1500)is artificially excluded from the basis below 1.7 GeVand the wave with thef₀(980)is only added to the basis above 1.3 GeV.

As we mention above, the symmetrization introduces a non-orthogonality in the basis. It causes a

“leakage” between waves. An example of this issue is shown in Fig. 5.9: S= 3is not part of our Deck model, however, theJ^{P C}M = 2⁻⁺0⁺ρ₃π P-wave gets non-zero weight, when the symmetrized amplitude is expanded in the symmetrized basis.

Im Dokument Three-pion dynamics at COMPASS: (Seite 94-99)