Recent progress in the partial-wave analysis of the diffractively produced π − π + π − final state at COMPASS
RRecent progress in the partial-wave analysis of the diffrac-tively produced π − π + π − final state at C ompass Fabian Krinner , ∗ for the C ompass Collaboration Technische Universität München, Munich, Germany
Abstract.
The C ompass spectrometer at CERN has collected a large data set fordi ff ractive three-pion production of 46 × exclusive events. Based on previ-ous conventional Partial-Wave Analyses (PWA), we performed a “freed-isobarPWA” on the same data, removing model assumptions on the dynamic isobaramplitudes for dominating waves. In this analysis, we encountered continuousmathematical ambiguities, which we were able to identify and resolve. Thisanalysis gives an unprecedented insight in the interplay of 2 π and 3 π dynamicsin the process. As an example we show results for a spin-exotic wave J PCX − = − + wave. π production at C ompass In this work, we perform a Partial-Wave Analysis (PWA) of the di ff ractive process π − beam + p target → X − + p recoil → π − π + π − + p recoil measured by the C ompass experiment at CERN using an 190 GeV / c negative hadron beam—consisting to 97% of negative pions—impinging on a liquid hydrogen target. For this processthe C ompass collaboration has recorded a very large data set of 46 × exclusive events,performed a very detailed PWA on this data using a set of 88 partial waves [1], and extractedresonance masses and widths of eleven intermediate isovector resonances X − [2]. The PWA preformed in Ref. [1] relies on the isobar model, in which the decay of the interme-diate state X − into the 3 π final state is modeled as a sequence of two-particle decays involvinga second intermediate state ξ , the isobar: X − → ξ + π − bachelor → π − π + π − ;the best known examples for such isobars are the ρ (770), f (980), and f (1270) resonances.In the conventional approach, the dynamic amplitudes of these isobar resonances ∆ ξ (cid:0) m ξ (cid:1) —often called “line shape”—are a necessary input for the PWA model. The most commonexample for such a dynamic isobar amplitude is the well-known Breit-Wigner amplitude withgiven resonance mass and width. However, the necessity for fixed dynamic isobar amplitudesin a conventional PWA hast several disadvantages, since ∗ e-mail: [email protected] a r X i v : . [ h e p - e x ] A ug it is not a priori clear, which isobar resonances to include in the analysis model, • Breit-Wigner amplitudes may not give an accurate description of all isobars, • and overlapping Breit-Wigner amplitudes violate theoretical requirements.To avoid these drawbacks of the conventional approach, we use an analysis technique called“freed-isobar PWA”—also often named “model-independent PWA”—where we replace thefixed dynamic isobar amplitudes by sets of bin-wise constant functions: ∆ ξ (cid:0) m ξ (cid:1) → (cid:88) bins ∆ bin ξ (cid:0) m ξ (cid:1) with ∆ bin ξ (cid:0) m ξ (cid:1) = m ξ in the bin , m ξ bin behaves like an independent partial wave in the analysis modelthe freed-isobar approach allows to re-used the existing analysis scheme with a much highernumber of degrees of freedom. Bin-wise approximations to the dynamic isobar amplitudesare hereby encoded in the strengths and relative phases of these individual partial waves. Thisapproach allows to resolve the process in terms of the angular-momentum quantum numbersand the mass of the isobar. Partial waves with dynamic isobar amplitudes replaced this waywill be called “freed” from hereon.We performed such a freed-isobar analysis on the data set introduced in Sect. 1 using thesame wave-set and freeing the following 12 of the total 88 waves:0 − + + [ ππ ] ++ π S , ++ + [ ππ ] ++ π P , − + + [ ππ ] ++ π D , − + + [ ππ ] ++ π S , − + + [ ππ ] −− π P , ++ + [ ππ ] −− π S , − + + [ ππ ] −− π P , − + + [ ππ ] −− π P , − + + [ ππ ] −− π P , ++ + [ ππ ] −− π S , − + + [ ππ ] −− π F , ++ + [ ππ ] −− π D . The freed waves were chosen to be the 11 waves with the highest intensity in the conventionalanalysis plus the spin-exotic J PCX − = − + wave, which is a wave of major interest. Since partialwaves with identical angular-momentum quantum numbers are absorbed in a single freedwave, this leaves 72 waves with fixed dynamic isobar amplitudes in the model.The analysis was performed in 50 independent bins in the invariant mass of the 3 π system,from 0 . . / c and four non-equidistant bins in the four-momentum transfer t (cid:48) in theanalyzed region from 0 . . / c ) , giving a total of 200 independent fits. The widthof the m ξ bins was chosen to be 40 MeV / c , with smaller widths in the regions of knownresonances: 20 MeV / c in the regions of the ρ (770) and the f (1270), and 10 MeV / c in theregion of the f (980). The fact, that models in a freed-isobar PWA have a much higher number of degrees of free-dom, may lead to the appearance of continuous ambiguities, caused by exact cancellationsbetween di ff erent terms of the amplitude. Such cancellations, which we call “zero modes”from hereon, therefore have to be identified and the corresponding ambiguities have to beresolved.Since this article focuses on the spin-exotic wave, we show how a zero mode arises withinthis particular wave. The dependence on the kinematic variables of the decay X − → π − π + π − of the spin-exotic wave is determined by its angular-momentum quantum numbers and givenby: ˆ A (12)1 − + ∝ ( (cid:126) p × (cid:126) p ) ∆ − + (cid:0) m (12) ξ (cid:1) , (2)here the appearing three-momenta are defined in a rest system of X − and we have assumedthat the isobar is formed by π − and π + . However, since there are two identical π − in the final-state, the amplitude has to be Bose symmetrized and the total amplitude of the spin-exoticwave is: A − + = ˆ A (12)1 − + + ˆ A (23)1 − + ∝ ( (cid:126) p × (cid:126) p ) (cid:20) ∆ − + (cid:0) m (12) ξ (cid:1) − ∆ − + (cid:0) m (23) ξ (cid:1)(cid:21) , (3)where the respective minus sign stems from the exchange of π − and π − in Bose symmetriza-tion and the antisymmetry of the cross product. From this equation, we can easily see, that theamplitude of the spin-exotic wave is invariant under a change of the dynamic isobar amplitudeby: ∆ − + (cid:0) m ξ (cid:1) → ∆ − + (cid:0) m ξ (cid:1) + C , (4)since both Bose-symmetrization terms exactly cancel in Eq. (3). Therefore, also the intensityand the likelihood function are invariant under this transformation. Thus, we have identifieda zero-mode in the spin-exotic wave, where the corresponding ambiguity is encoded by thecomplex-valued coe ffi cient C .Since the likelihood function is invariant under a change of the zero-mode coe ffi cient C , the fitting algorithm may find a solution with any arbitrary value for it, which might notrepresent the physical one. Therefore, we have to adjust C in a second fit step, using additionalconditions on the resulting dynamic isobar amplitude as constraint. In the case of the spin-exotic wave, we required the resulting dynamic isobar amplitude to be as close as possibleto a Breit-Wigner shape for the dominating ρ (770) resonance within the scope of the soleparameter C . To minimize possible e ff ects of excited ρ (cid:48) resonances, we limited the fit rangeto isobar masses below 1 .
12 GeV / c .Note, that this second fit step fixes only a single complex-valued degree of freedom, while n bins − With the model defined in Sec. 2 and the method to resolve the zero-mode ambiguity givenin the previous section, we can analyze the data-set of Sec. 1 and obtain an unprecedentedinsight into the dynamics of di ff ractive three-pion production.The result for a single ( m π , t (cid:48) ) bin is shown in Fig.1, we see, that the resulting dynamicisobar amplitude is dominated by the ρ (770) resonance, as expected. The fixed Breit-Wigneramplitude is a good approximation to the dynamic isobar amplitude, with some significantdeviations, especially in the peak region. Such deviations might be caused by non-resonantcontributions to the process or re-scattering e ff ects with the third pion.Looking at the dependence of the freed-isobar results on the m π bin, as shown in Fig. 2,we find a nice correlation of the intensity distributions in m π and m ξ , which correspondsto the dominating decay π − (1600) → ρ (770) π − . The coherent sum of all m ξ has a similarpeak position and width, as the result of the conventional analysis in Ref. [1], but a highertotal intensity. This shows that the π − (1600) resonance found in the conventional analysis ofRef. [2] is not an artifact of the fixed dynamic isobar amplitudes used. We performed an extended freed-isobar PWA for di ff ractive 3 π production, for which theC ompass spectrometer has collected large exclusive data set of 46 × events. In this ana- . . m π − π + [GeV / c ] I n t e n s it y [ E v e n t s / ( G e V / c )] × − + + [ ππ ] −− π P1 . < m π < . / c . < t < . / c ) P r e li m i n a r y Corrected zero modeFixed shape Full range − Re( T bin )[(Events / (GeV / c )) / ] × I m ( T b i n )[( E v e n t s / ( G e V / c )) / ] × − + + [ ππ ] −− π P1 . < m π < . / c . < t < . / c ) P r e li m i n a r y Corrected zero modeFixed shape Full range
Figure 1.
Results for the dynamic isobar amplitude of the spin-exotic wave for a single bin in m π and t (cid:48) in two di ff erent representations. The resulting intensity distribution is shown on the left, and the Arganddiagram on the right. The gray line indicates the constraint used to resolve the zero-mode ambiguity. . . . . . m π [GeV / c ] I n t e n s it y [ E v e n t s / M e V ] × − + + [ ππ ] −− π P0 . < t < . / c ) P r e li m i n a r y Fixed isobars Freed isobars . . . . . m π [GeV / c ] . . . . m π [ G e V / c ] − + + [ ππ ] −− π PPreliminary0 . < t < . / c ) Figure 2.
Results of the freed-isobar analysis of the spin-exotic wave as a function of m π . The two-dimensional intensity distribution is shown on the right and the coherent sum of all m ξ bins on the left,compared to the results of the conventional PWA. lysis, we encountered continuous mathematical ambiguities—zero modes—, which are iden-tified and resolved. The results for the spin-exotic wave showed, that the dynamic isobaramplitude is dominated by the ρ (770) resonance, as expected. However, some small but sig-nificant deviations from a pure Breit-Wigner shape are visible. We compare our findings tothe conventional PWA method and find a peak compatible with the π − (1600) resonance ofRefs. [1, 2]. References [1] C ompass collaboration (C. Adolph et al. ), Phys.Rev.
D95 (2017) no.3, 032004[2] C ompass collaboration (R. Akhunzyanov et al. ), arXiv:1802.05913 [hep-ex] (2018)3] F. Krinner, D. Greenwald, D. Ryabchikov, B. Grube, and S. Paul,Phys. Rev.