The studies of light-meson resonances are the main subject of this thesis. The central issue of this research is that the interaction between hadrons cannot be calculated from first principles. Although QCD is known and accepted to be the fundamental underlying theory of the strong interaction, the consequences for the hadron spectrum cannot be inferred from it due to the complicated, non-perturbative nature of this theory. Among several possible phenomenological treatments of hadron spectroscopy, we attempt to pursue the most general approach which is based on basic properties of the scattering amplitude: unitarity, analyticity and crossing symmetry. The approach allows us to strictly define the resonance characteristics such as the mass and width and remove a large arbitrariness of the phenomenological amplitude construction in other approaches.
The work is focused on the dynamics in the system of three charged pions (π−π+π−) from measurements of diffractive reactionsπ−p→π−π+π−p, performed at the COMPASS experiment.
In Sec. 3 we have introduce the main featured of the three-pion physics seen in the diffractive production: The three-pion interaction incorporates numerous different excitations modes: the system couples to a few tens of hadronic states that arise either as excitation of a subsystem (two pions) or the excitation of the system as a whole. The identification, separation, and characterization of the excited mesonic states is a very challenging problem that has been investigated by physicists around the world for many years. Moreover, in addition to the hadronic excitations, the physics of the strong interaction exhibits the quantum phenomena of final-state. Attempting to advance the customary approach of modeling the energy dependence of the amplitude by several Breit-Wigner terms and a phenomenological background, we have performed exploratory studies in both theoretical and practical directions of research.
We have studied the origin of the new resonance-like phenomenona1(1420), observed by the COMPASS experiment in 2015 in thef0π P-wave. We have suggested and developed an interpretation of the signals as a consequence of a peculiar3π-KKπ¯ coupling. A close-to-mass-shell kaon exchange, whenK∗K¯ scatters tof0π, has been found to produce an enhancement in the spectrum off0πand a significant motion of the scattering phase. All features of the phenomenon established by the COMPASS data have been reproduced within a simple model (see Chapter 4). The understanding of the rescattering effect might shine light to theXY Zstates discovered in the charmonium sector as well as the pentaquark states observed by LHCb. There are indications that some of these exotic-candidate signal might originate from rescattering effects. In this sense, thea1(1420)signal is a gift: due to the rather well-known interaction between light hadrons and thanks to high statistical precision of modern experiments, we have a chance to illuminate this tricky feature of the strong interaction.
In addition to rescattering effects, we have also studied a second long-standing issue in spectroscopy:
how to formulate amplitudes in a theoretically cleaner way than the traditional approach of a sum of the Breit-Wigner amplitude. Starting from the well-established unitarity-based technique (see Chapter 2) for a system of two particles, we approach the3πsystem where two pions are combined
into a quasi-stable intermediate state, e.g. the ρ-meson, or the f2-meson. Since the diffractive reactions exhibit complicated production mechanisms including non-resonant background processes, we have applied our model to the hadronic decays of theτ-lepton,i.e. τ →3π ντ. Using the publicly available data of the ALEPH experiment, we succeeded to fix the model parameters and to perform an analysis of the analytic structure of the scattering amplitude (see Chapter 6). The pole position of the spin-partner of theρ-meson, the ground axial vectora1(1260)has been extracted for the first time.
Using the ALEPH data we have tested our model which included the interference of two interaction chains (ρ[→π1−π2+]π3−andπ1−ρ[→π+2π3−]) against the simpler one of the quasi-stableρ-meson and a bachelor pion resonating in theS-wave. The comparison has revealed a significant difference in the results of the two models; it has emphasized the importance of the fact that the3π-system is not just a quasi-two-particle system.
A further important achievement of this thesis is the realization of a coherent framework which puts both the final-state interactions and the three-particle resonances to the same footing (Chapter 7).
Using a simplified version of the3π system (three identical scalar particles) we have derived the three-body unitarity requirements and suggested a model which manifestly satisfies them. We show that for a system with a significant subchannel interaction, the rescattering processes generate the long-range interaction amplitude which is a sum of ladder-like diagrams (the two-particle resonances in different subsystems are formed and decay via one-particle exchange). The three-particle resonances are incorporated into the model by short-range terms (e.g. contact terms). In order to preserve three-body unitarity, these terms need to be “dressed” with the ladder rescattering. A dedicated publication is being drafted.
The framework unifies several approaches applied to a system of three particles: in the limit of a stable subchannel resonance, it reduces to simple two-body unitarity-based constructions (e.g. the K-matrix); under a slightly weaker approximation, the framework leads to the approach used in theτ-decay analysis mentioned above. For a fixed value of the total invariant mass, the model for the subchannel dynamics of our general framework is reminiscent of the Khuri-Treiman model (see Sec. 4.5 and Sec. 7.2). It follows that the three-hadron scattering amplitude can be written and the N-over-Dfunction (see Eq. (7.35) and Eq. (7.36) withN =k(1 +τL0)and1/D=R) where the N includes the details of the production and decay chains, while theDdescribe the three-particle resonances (bare poles dressed by the hadronic loops). The final-state interaction modifies the line shape of theππ-subchannel resonances; it influences the results for the pole position of three-particle resonances (as we also found in the systematic studies of theτ-decay analysis in Sec. 6.4).
The framework is well suited for addressing thea1(1420)phenomenon, although it would require an extension to accommodate two coupled channels (3πandKKπ). The main question concerning¯ the nature of thea1(1420) is either if the rescattering effects incorporated by theN function are sufficient to describe the data, or if the functionDrequires (develops) poles in addition to the expected axial states. It is important to notice, that the triangle singularity from the kaon exchange is present in theN-function as a part for the decay process. An investigation using this approach is ongoing.
Along the path of the investigation on thea1(1420), an essential next step would be a comprehensive analysis of the decay τ → 3π ντ. A simple test of the a1(1420) based on the dissection check discussed in Sec. 4.1 (see Fig. 4.3) can be done on theτ-sample ahead of thePWA. The conventional PWA(see Sec. 3.3.1) might be used to confirm the COMPASS observation of thea1(1420). The freed-isobarPWAdiscussed in Sec. 3.4 would give a model-independent extraction of theIsobarline shapes (π+π−interaction withS- andP-waves in this case), distorted by the final-state interaction.
Sequentially, these results have to be subjected to the dispersive unitarity-based approach such as the
7.3 Conclusions
Khuri-Treiman model. Subtraction constants need to be fitted for every fixed value of the3πinvariant mass. The final step of the “comprehensive analysis” is a dynamic description of the subtraction constants as functions ofm3π, using the general framework developed in Chapter 7 and the analytic continuation of the model to the complex plane. We admit that the project is ambitious: it would require a lot of research as well as large computational resources for every step of the sketched plan. It is not less important to have a sufficiently large and precise data sample. Belle and BaBar are the two experiments which have collected over108τ-pair events produced ine+e−collisions [229, 230].
Studies of the 3π system in diffractive reactions are more complicated due to the production mechanism. However, over the last years, COMPASS performed an outstanding work for breaking down this complex data into separate intensities for differentJP C sectors. With the recent result of the freed-isobar analysis, one gets access to the subchannel spectra distorted by the rescattering effects. Those results seems to be the best “data” to validate our understanding of the rescattering physics. For example, the exotic sector withJP C = 1−+is found to couple exclusively (among the 3πwaves in the COMPASSPWA) to theρπ P-wave. The Khuri-Treiman model in this case resembles the model forω→π+π−π0studied in Ref. [165, 166] with a slight difference due to the total isospin
6(an investigation has been started in collaboration with Tobias Isken and Bastian Kubis). We have performed the first steps in generalizing the KT-formalism for arbitraryJP C quantum numbers of the system. The application of this approach toJP C = 1++based on the COMPASS freed-isobar results is an ongoing JPAC project.
Combining the rescattering studies with the three-body resonance extraction is, however, difficult given the COMPASS data. An additional challenge with respect to theτ analysis is the non-resonant background process, the Deck process. The understanding of its effect on the three-particle spectrum is still far from being quantitative. everal partial waves in the COMPASS analysis [78] are believed to contain a significant contribution of this background. In Chapter 5 we have consider available models for the Deck processes and sorted out an important parameters. We have found that the line shapes of the partial-wave projections vary substantially depending on a model for the exchange-pion propagator.
An adjustment of the background parametrization using the COMPASS data is the next mandatory step to move forward the understanding of diffractive reactions (this is an ongoing work).
An investigation of alternative production mechanisms, such ase.g. γ p→3π p, that are possible at the CLAS12 and GlueX experiments, will provide complementary access to the meson excitation spectrum. Interestingly, the CLAS partial-wave analysis of Ref. [231] reported no evidence for the exoticπ1(1600) signal inJP C = 1−+ as well as no evidence for thea1(1260)in theJP C = 1++
sector. In this respect, the data from the GlueX experiment are looked forward to [232]. The techniques and methods developed in this thesis will be essential in the future analyses.
6Essentially, the inhomogeneous term in the KT framework contains twoππcross channels in case ofω-decay:JP C = 1−−, the total isospin isI= 0. For the3πsystem withJP C = 1−+,I= 1, there are twoπ+π−pairs.