Preliminary

] c2

[GeV/

π m3

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 )2cIntensity / (20 MeV/

10 20 30 40 50

103

× π⁻p→π⁻π⁻π⁺p (COMPASS 2008) πS

(1270) f2

0+ +

2−

)2

c < 0.113 (GeV/

t' 0.100 <

Mass-independent fit Mass-dependent fit resonant non-resonant

A.2 Mass-dependent fit of the2⁻⁺sector of the helicity non-flip amplitude. The functionZ_LS^JM is defined in Eq. (3.9) with the reflectivity index introduced in Eq. (3.11). The isospin coefficientC_I is given in Eq. (3.8).

Simplification 3: We assume a factorization and a simpleσdependence ofF_LS^JM .

F_LS^JM (s, σ) =FLS^JM (s)h_L(s, σ)f_S(σ), (A.9) whereFLS^J (s)is an amputated dynamic production amplitude which does not depend onσ. The functionf_S(σ)is the isobar line shape as shown in Appendix D. Theh_L(s, σ)is a factor which ensures a correct threshold behavior and asymptotic limit. We use a simplified form of the Blatt-Weisskopf factors,

h²_L=

R²q² 1 +R²q²

L

, q = 4πρ_S√

s. (A.10)

In order to use the unitarity constraint we need to introduce the3→3scattering amplitude (see Sec. 7), which, under assumptions we made, reduces toξπ → ξ⁰π scattering. In order to reduce the general model discussed in Sec. 7 to the quasi-two-body scattering, theladder interactionhas to be neglected. The full scattering amplitudeT is parametrized by theshort-rangeinteraction. The partial-wave expansion ofT reads,

T =X

J

X

L⁰S⁰LS

X

M

Z_LS^{JM ∗}(Ω⁰₁,Ω⁰₂₃)T_L^J0S⁰LS(σ₁, s, σ₂₃)Z_L^JM 0S⁰(Ω⁰₁,Ω⁰₂₃), (A.11) where we consistently use the factorization assumption,

T_L^J0S⁰LS(σ⁰, s, σ) =f_S0(σ⁰)h_L0(s, σ⁰)T_L^J0_S⁰_LS(s)h_L(s, σ)f_S(σ). (A.12) The amplitudeT is independent of the production process. The relation to the production amplitude FLS^J (s), is obtained through the unitarity relation:

2 ImFLS^JM (s) =T_LSL^{∗J 0}_S⁰(s) Σ_L0S⁰(s)F_L^{JM 0}_S⁰(s), (A.13) 2 ImT_LSL^{J 0}_S⁰(s) =T_LSL^{∗J 00}_S⁰⁰(s) Σ_L00S⁰⁰(s)T_L^J00_S⁰⁰_L⁰_S⁰(s), (A.14) where the quasi-two-body phase-space factor reads,

Σ_LS(s) = 1 (8π)²

Z ₍^√_s−mπ)²

4m²_π

dσ

2πh²_L(s, σ)λ^1/2(s, σ, m²_π)λ^1/2(σ, m²_π, m²_π)

sσ |f_S(σ)|². (A.15)

Parametrization of the scattering matrix and the pole search

The unitarity constraint to the scattering amplitudeT in Eq. (A.14) can be built in using theK-matrix approach (see Sec. 2.2). In matrix notations overLSindices (J is fixed to2), it reads,

T(s) =K(s)h

1−iΣ(s)K(s)/2˜ i₋1

, (A.16)

whereK(s)is a matrix of functions which is parametrized by the sum of pole terms,Σ˜ is a diagonal matrix with elementsρ˜_LS, that are given by a dispersive integral (cf. Eq. (A.17)),

Σ˜_LS(s) = 1 πi

Z _∞

s_th

Σ_LS(s⁰)

s⁰−s ds⁰, K_LSL0S⁰(s) =X

R

g^R_LSg_L^R0S⁰

s_R−s . (A.17)

The amplitudeT, is an analytic function, defined in the complexs-plane. Resonances in theξπ system are found by identifying poles in the unphysical Riemann sheets. The physical unitarity cut starts at the three pion threshold,√

s= 3m_π. The sheet attached to the real axis from below is called thesecond Riemann sheet. Every quasi-two body channel ofT introduces an additional cut which starts in the complex plane as demonstrated in details in Sec. 6.3.1. All those complex cuts need to be rotated (see Fig. 6.4) to give access to the closest unphysical sheet.

The scattering amplitude at the second Riemann sheet,T_II⁻¹, is calculated according to Eq. (2.32), whereΣ(s)used instead of the phase-space factorρ(s). Instead of looking for poles inT_II, we explore the inverse functionT_II⁻¹, and find the point where it vanishes exactly. It is practically convenient to search for the solution of the equation,det(T_II⁻¹K) = 0, which is equivalent to

det

1 +i( ˜Σ(s)/2 + Σ(s))K(s)

= 0. (A.18)

Candidates for the solutions are found by minimization of the modulus of the left part. The minima are validated by a direct check of Eq. (A.18).

Two models of the production amplitude

Two production models, introduced in Sec. 2.3, are considered

• Model−I: production vector.

Theproductionamplitude is written as the scattering matrix multiplied by an arbitrary source vector, denoted byα^JM _LS (s),

FLS^JM (s) =X

L⁰S⁰

T_LSL^{J 0}_S⁰(s)α_L^JM 0S⁰(s). (A.19) The functionsα^JM _LS (s), cannot have the a right-hand cut in order to preserve unitary. To model these functions, polynomial series of the conformal variableω(s), are used.

α_w(s) =e^iφwX

α⁽ⁱ⁾_w ωⁱ(s), ω(s) =

√s₁−√s+s₀

√s₁+√

s+s₀, (A.20) where we used a combined indexw= (JM LS)for brevity of notation. The variabless₀ and s₁ are parameters of the conformal map: s₀ gives the position of the left-hand cut, whiles₁ determines the slope of thesdependence. The expansion coefficientsα_w⁽ⁱ⁾, are real, a single complex phaseφ_w, is a free parameter of the fit. For indication of the parameter space, we introduce a superscript. Model−I^(n,m,l)refers then-channels amplitude withmpoles in the K-matrix andlterms in the polynomial series.

A.2 Mass-dependent fit of the2⁻⁺sector

• Model−II: unitarized background.

The second model uses an explicit background termB^JM LS (s), which is incorporated according to Eq. (2.41).

FLS^JM (s) =B^JM LS (s) +X

L⁰S⁰

T_LSL^{J 0}_S⁰ c^JM _L0S⁰ + 1 2π

Z _∞

s_th

B^JM _L⁰_S⁰(s⁰) Σ_L0S⁰(s⁰) s⁰−s ds⁰

!

, (A.21) wherec^JM _L0S⁰ parametrizes the short-range interaction. These coefficientsc, are complex constants adjusted in the fit. Eq. (A.21) resembles the model suggested by Basdevant and Berger in Ref. [62].

The background BLS^JM (s), is calculated by projecting the “Standard Deck” amplitude (see Sec. 5) to the partial waves. The projections, however, haveσdependence which cannot be addressed with the approximations stated above. An artificial method is used to get rid of this dependence. The subchannel invariant mass in those projections is set to the nominal mass of the isobar above the isobar-pion threshold,√s_th =m_ξ+m_π. Below the√s_th, the projections are evaluated for the value ofσthat are on the interpolation between the nominamIsobarmass and the two-pion threshold,√σ_th= 2m_π. This way of omitting theσdependence makes the background model unreliable (see the discussion on features of the projections in the COMPASS basis in Sec. 5.4). Moreover, there are large uncertainties on the line shape of the partial wave projections due to questions on the dynamic model for the pion propagator discussed in Sec. 5.3.

Nevertheless, the obtained projections have two simple features of the background observed in the data. It rises near thresholds_th, and falls at high energies. The projections are purely real. Due to this large uncertainty, the relative strength of the background was not fixed. The background couplings are free parameters for all waves and allt⁰slices. Model−II^(n,m)refers then-channels amplitude withmpoles inK-matrix and the unitarized background.

Fit to the COMPASS data and extracted poles

Free parameters of the models are positions of theK-matrix poles, the expansion coefficients of the production coefficients inModel−I and the strength of the “background” term and the “direct production” term in theModel−II. The data points from the COMPASSPWAare given in slices of t⁰. The differences in the intensity distributions for differentt⁰slices, are attributed to the variations in the production mechanism. All data have to be fitted simultaneously using an independent set of production parameters for everyt⁰-slice and one production-independent set of parameters for the scattering matrix.

To address the feasibility of the approach, many exploratory studies have been performed. The Model−I was found very flexible due to the large freedom of adjusting the production parameters (the model typically contains a few hundred free parameters). Due to the same reason, the model is found to have stability problems. Many local minima are barely distinguishable, and the fit parameters are largely correlated. Fig. A.3 and Fig. A.4 show the firstt⁰-slice of the fit withModel−I^(3,3,6)and Model−I^(4,4,7), respectively. The plots demonstrate that the intensities can be described rather well within theModel−I. Most of the features seen in the data are reproduced in the fit. Namely, both the main peak and the shoulder of thef₂π S-wave are described. The structure at1.8 GeVappears differently in different waves: it is a dip in the(ππ)_Sπ D-wave and a well pronounced peak in the

Figure A.3: The results of the fit of the dynamic modelModel−I^(3,3,6)to the COMPASSSDMwithJ^{P C}M= 2⁻⁺0⁺. The fit is performed simultaneously with all11t⁰slices, the first slice,0.1< t⁰<0.112853 GeV², is shown. The columns and rows are numbered from left to right and from top to bottom: they refer to the following waves:f2π S-wave,(ππ)Sπ D-wave, andρπ F-wave. The wave intensities are shown on the diagonal of the plot matrix. The off-diagonal elements are the real (upper triangle) and imaginary (lower triangle) parts of the interference terms (see off-diagonal elements of theSDMin Eq. (3.25)).y-axis shows intensity of the waves and the interference terms expresses in the number of events,x-axis is the invariant mass of the three-pion system in GeV. The model includes three poles in theK-matrix and five expansion terms in the production vector for every channel. The other colored lines illustrate projections of the model to the intensities and interference terms when all but oneK-matrix poles are turned off (couplings in Eq. (2.24) are set to zero). The range where the model has been fitted is indicated by solid lines, the dashed parts show the extrapolation of the model.

f₂π D-wave as shown in Fig. A.6. The three hills of theρπ F-wave are reproduced in both models, however, we notice that the contribution of theK-matrix parameters is quite different for the two models,Model−I^(3,3,6)andModel−I^(4,4,7). Parameters of theK-matrix are not physical, and only the poles ofT do identify hadronic resonances. However, there is an asymptotic correspondence between theK-matrix poles and the poles ofT-matrix. When theK-matrix couplings in Eq. (2.24) are scaled simultaneously to zero, theT-matrix approachesK as seen in Eq. (A.16) (the inverted matrix in Eq. (A.16) is an identity in the limit). Therefore, for every pole which is put in theK-matrix, there is the corresponding pole in the complex plane.

The analytic continuation is presented in the Fig. A.5 forModel−I^(3,3,6), and Fig. A.6 for the Model−I^(4,4,7). In both attempts, three poles of theK-matrix have converged to values close to the range of the fit. The fourthK-matrix pole in Fig. A.6 ended up at higher energy outside of the fit

A.2 Mass-dependent fit of the2⁻⁺sector

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0

5000 10000 15000 20000 25000 30000 35000 40000

45000 π S-wave

2 Intensity of f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−4000

−2000 0 2000 4000

6000 π S-wave f₂π D-wave

2 Real part of interference f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−2000 0 2000 4000 6000 8000 10000 12000 14000

16000 π S-wave ρπ F-wave

2 Real part of interference f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−2000 0 2000 4000 6000 8000 10000 12000 14000 16000

18000 _D-wave

S π) π S-wave ( π 2 Real part of interference f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−2000 0 2000 4000 6000 8000

10000 π S-wave f₂π D-wave

2 Imag part of interference f

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0

1000 2000 3000 4000

5000 π D-wave

2 Intensity of f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−1000 0 1000 2000 3000 4000

F-wave π ρ D-wave π 2 Real part of interference f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−3000

−2000

−1000 0 1000 2000 3000

4000 D-wave

S π) π D-wave ( π 2 Real part of interference f

1 1.2 1.4 1.6 1.8 2 2.2 2.4 0

2000 4000 6000 8000 10000

12000 Imag part of interference f₂π S-wave ρπ F-wave

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−5000

−4000

−3000

−2000

−1000 0

F-wave π ρ D-wave 2π Imag part of interference f

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0

5000 10000 15000 20000 25000 30000

35000 Intensity of ρπ F-wave

1 1.2 1.4 1.6 1.8 2 2.2 2.4 0

1000 2000 3000 4000 5000 6000 7000 8000

9000 D-wave

S π) π F-wave ( π ρ Real part of interference

1 1.2 1.4 1.6 1.8 2 2.2 2.4 0

2000 4000 6000 8000 10000

12000 D-wave

S π) π S-wave ( 2π Imag part of interference f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−2500

−2000

−1500

−1000

−500 0 500

D-wave S π) π D-wave ( 2π Imag part of interference f

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−3500

−3000

−2500

−2000

−1500

−1000

−500 0 500

1000 D-wave

S π) π F-wave ( π ρ Imag part of interference

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0

2000 4000 6000 8000

10000 D-wave

S π) π Intensity of (

Figure A.4: Same as Fig. A.3 but with theModel−I^(4,4,7). The used waves aref2π S-wave,f2π D-wave, ρπ F-wave and(ππ)Sπ D-wave. y-axis shows intensity of the waves and the interference terms expresses in the number of events,x-axis is the invariant mass of the three-pion system in GeV.

range. We find three resonance poles, which would correspond toπ₂(1670),π₂(1880), andπ₂(2005).

The positions of those poles, however, are not well determined.

The approach used inModel−IIhas much fewer parameters, the minimum is better determined. On the other hand, it gives worse description of the data. Especially, the description of theρπ P-wave and f₂π D-wave was found very problematic, possibly because these waves require a large background fraction. The largest set of the data which is reasonably described within theModel−IIis the three waves set presented in Fig. A.7. The main problem of this approach seems to be the simplistic and possibly incorrect background model. It is indirectly seen from the fact that the amplitude for the background, drawn by a light-green line in Fig. A.7, is nearly zero forf₂π S-wave and theρπ F-wave.

It seems to be preferred by the fit procedure, to assign the whole amplitude to the short-range component (“direct production”), because this part is more flexible, while the used background does not necessarily represent the physics right. The resonant poles found for theModel−II^(3,3)are seen in Fig. A.8. The fit seems to prefer a narrowπ₂(1880)and a slightly lighterπ₂(1670). The distant π₂(2005)does not look reliable. An isolated contribution of this pole is seen on the intensity plot for the(ππ)_Sπ D-wave. Its interference to the other components and the background is important, therefore, large correlations to the background shape are expected.

The results of the three models highlighted in the text are summarized in Table A.3. The poles are ordered by their mass and assigned toπ₂(1670),π₂(1880), andπ₂(2005). In the conventional

-1K]

Ln@Abs part of det[TI,II

] s M=Re[

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

]s=2Im[Γ

−1

−0.5 0 0.5 1

-1K]

Ln@Abs part of det[TI,II

Figure A.5: Analytic continuation of the scattering amplitude inModel−I^(3,3,6)to the unphysical sheet. The expression|det(T⁻¹K)|is calculated in the complexsplane at the pointss= (M −iΓ/2)², whereM in the units of GeV is given by thexcoordinate, andΓin the units of GeV is set by theycoordinate. For the positive values ofΓ(y-axis), the expression|det(T(s)⁻¹K(s))|is presented by the color code, while for the negative values ofΓ(nagative values alongy-axis), the values of the expression|det(TII(s)⁻¹K(s))|are presented.

The red spots in the plot are zeros of the plotted expression. Those are positions of theK-matrix poles. The blue spots are resonance poles.

Table A.3: A summary of the limited systematic studies on the pole positions of theπ₂resonances are shown in black. The values state the limits for the pole parameters, they are found by comparing the three models presented in the text: Model−I^(3,3,6),Model−I^(4,4,7), andModel−II^(3,3). Those number can be compared to theBreit-Wignerparameters obtained in the mass-dependent fit of Ref. [3] which are stated in gray. The pole positions for these parametrization are discussed in Sec. 3.3.3.

m_p,MeV Γ_p,MeV m_BW,MeV Γ_BW,MeV π₂(1670) 1640. . .1720 250. . .320 1642⁺¹²₋₁ 311⁺¹²₋₂₃ π₂(1880) 1810. . .1890 150. . .220 1847⁺²⁰₋₃ 246⁺³³₋₂₈ π₂(2005) 1950. . .2350 850. . .1050 1962⁺¹⁷₋₂₉ 269⁺¹⁶₋₁₂₀

approach of Ref. [3] the amplitude is modeled by a sum ofBreit-Wigneramplitudes. In Sec. 3.3.3 we showed that the pole positions estimated from theBreit-Wignerparameters are close to these values,m_BWandΓ_BW. It makes it reasonable to compare it to our results from unitary based models.

Our results have much large systematic uncertainties. The parameters ofπ₂(1670)agree within their uncertainties, while we observe narrowerπ₂(1880)and much widerπ₂(2005).

A.2 Mass-dependent fit of the2⁻⁺sector

-1K]

Ln@Abs part of det[TI,II

] s M=Re[

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

]s=2Im[Γ

−1

−0.5 0 0.5 1

-1K]

Ln@Abs part of det[TI,II

Figure A.6: Same as Fig. A.5 but with theModel−I^(4,4,7).M andΓare the units of GeV.

Summary

We have performed fusibility studies of the unitarity based approach in the mass-dependent fit of the COMPASS data. We made a set of approximations which were required by the current model of the PWA: the final-state interaction was neglected in the production amplitude; we also neglected the ladder interaction in the scattering amplitude. The analysis was performed under the assumption of quasi-stable isobars.

We were able to describe a subset of2⁻⁺waves from the COMPASSPWAwith a quality similar to the conventionalBreit-Wigneranalysis of Ref. [3]. Using methods of analytic continuation we extracted the pole position of theπ₂ states. In agreement with the current understanding of the excitedπ₂states, we found three resonances. We encounter several problems with the fit stability and multimodality of the solutions. In our model, the scattering isobar pion amplitude is well constraint, however, due to the unknown details of the production mechanism, the obtained pole positions have large systematic uncertainties. The preliminary results look promising and indicate a strong need for a better understanding of the background processes.

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

0 5000 10000 15000 20000 25000

30000 Intensity of f₂π S-wave

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−2000 0 2000 4000 6000 8000 10000

12000 π S-wave ρπ F-wave

2 Real part of interference f

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−2000 0 2000 4000 6000 8000

10000 D-wave

S

π) π S-wave (

2π Real part of interference f

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−1000 0 1000 2000 3000 4000 5000 6000 7000

8000 π S-wave ρπ F-wave

2 Imag part of interference f

π

M3

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

0 5000 10000 15000

20000 Intensity of ρπ F-wave

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

0 1000 2000 3000 4000

5000 D-wave

S

π) π F-wave ( π ρ Real part of interference

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−1000 0 1000 2000 3000 4000 5000 6000

D-wave

S

π) π S-wave (

2π Imag part of interference f

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

−3000

−2500

−2000

−1500

−1000

−500 0 500

1000 D-wave

S

π) π F-wave ( π ρ Imag part of interference

π

M3

1 1.2 1.4 1.6 1.8 2 2.2 2.4

0 1000 2000 3000 4000 5000

D-wave )S

π π Intensity of (

Figure A.7: Same as Fig. A.3 but with theModel−II^(3,3). The used waves aref₂π S-wave,ρπ F-wave and (ππ)Sπ D-wave. y-axis shows intensity of the waves and the interference terms expresses in the number of events,x-axis is the invariant mass of the three-pion system in GeV.

A.2 Mass-dependent fit of the2⁻⁺sector

-1K]

Ln@Abs part of det[TI,II

] s M=Re[

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

]s=2Im[Γ

−1

−0.5 0 0.5 1

-1K]

Ln@Abs part of det[TI,II

Figure A.8: Same as Fig. A.5 but with theModel−II^(3,3).M andΓare the units of GeV.

A P P E N D I X B

Im Dokument Three-pion dynamics at COMPASS: (Seite 158-169)