Overview

"You know, I couldn't do it. I couldn't reduce it to the freshman level.
That means we really don't understand it."
R.Feynman [Six Easy Pieces]

Strong interaction is one of the four fundamental interactions of nature -- in addition to Weak force, Electromagnetism, Gravity. Strongly interacting particles are called hadrons, with the most prominent representatives being protons, neutrons, pions, etc. . Hadrons make up most of the identified matter in the universe. Since the last century we know that most of hadrons are actually excited (short-lived) states. A reasonable analogy to this are the excitations of atoms (see visible spectrum of hydrogen below).

Model of the Hydrogen atom and its absorption/emission spectrum

The analogy to atomic system is a crude one: First, the scales of energies in hadronic systems are on the order of 1000 000 000 times larger than those of atomic systems; Second, in contrast to nucleus+electron(s) system the hadrons are constituent of not directly observable building blocks -- quarks and gluons (see Standard Model of Particle Physics). It is believed that understanding the emergence and pattern of the excited energy spectrum of hadrons will help us clarify the nature of strong interaction and, thus, the origin of mass/matter. For the current state of knowledge see the summary by the Particle Data Group. Theoretical understanding of hadronic spectrum has two indispensable ingredients -- quantum effects and (special) relativity. So far the only way to reconcile both is through the so-called Quantum Field Theory, which for the strong interaction is called Quantum Chromodynamics (QCD). This theory has several unusual properties rooted in its non-trivial mathematical structure. For example, protons are build up of three quarks, but have a mass hundred times larger than the sum of quark masses. Thus, the largest part of (ordinary matter) mass -- is due to QCD. Unraveling the pattern of excited hadrons is the golden thread through my research, connecting methods from S-matrix theory, effective field theories, numerical techniques of Lattice QCD, statistics, etc..



Resonances

Excited hadrons can manifest themselves in various observable reactions such, e.g., scattering or photoproduction experiments, through an excess in the cross-section rates. Using a concept of a periodically driven oscillator in classical mechanics and transferring it to quantum mechanics one can identify two properties of a resonant system -- position of the peak and its width (response of the system). It turns out that this identification is instructive in many cases, which however depends on the reaction type of the experiment. The most universal and modern approach to resonances deals directly with scattering amplitudes, relating universal properties of resonances (excited states) to the analytical properties of the transition amplitudes. In a nutshell, given an analytic form of the transition amplitude fixed by the experimental data for real energies, one extrapolates it to the complex energy plane. The latter so-called Riemann surface is in general a non-trivial manifold consisting of multiple (\(2^N\), if \(N\) two-body systems can be realized physically) Riemann sheets, each spanning over the whole complex energy plane. The first Riemann sheet is a pole-free surface, while the others can include poles (see Figure below).
Resonance poles on the second Riemann sheet in the complex energy-plane \(W_{\rm cms}\). Red line shows the part of the amplitude \(T\), which is constrained by the experimental data. [source]

Poles on the unphysical Riemann sheets are associated with the resonances, whose mass and width are exactly fixed by the complex-energy pole position. These both characteristics of an excited state are independent of the exact production setup. Besides different experimental setups this methodology in identifying universal parameters unites can also be applied to include theoretical constraints through, e.g., Effective field theories, Lattice QCD etc.. For more details see a recent dedicated review: One interesting state is the \( \Lambda(1405) \) resonance with quantum numbers \(I=0, S=-1\). It is important not only for testing of our understanding of low-energy QCD, but also in explaining novel states of matter searched for in J-PARC, GSI and DAFNE experimental facilities. It may, furthermore, be one of vehicles for explanation of the stiffness of the equation of state of neutron stars, thus, being of interest for applications in Astrophysics. For more details see a dedicated review: There are several other states of large phenomenological and theoretical interest which I have explored in the past using experimental scattering, photoproduction, electroproduction or lattice QCD results. Some of those states are parity partner of nucleon \(N(1535), N(1650)\), lightest unstable hadrons \(\rho(770), f_0(500)\), \(a_1(1260)\) (axial meson resonance decaying in \(3\pi\)), \(\Lambda(1405),\Lambda(1380)\Sigma(1385)\), ...:


Lattice QCD

Hadron spectroscopy directly from QCD is currently only accessible through the Lattice Gauge Theory. There, in large-scale numerical calculations, hadrons and interactions thereof can be studied in scenarios which may even be inaccessible by the experiments. Thus, Lattice QCD is more than a complementing tool to experiments but truly a theoretical approach to QCD.

Finite-volume effects
Quantization conditions

Lattice computations are necessarily performed on discretized Euclidean space time in a finite box. Both these technical limitations need to be lifted when comparing the results of Lattice QCD with phenomenological observations. The discretization effects are well under control, dealt with continuum extrapolations. Mathematically more intricate and interesting effects arise from the finiteness of the lattice setup -- the so-called finite-volume effects. At the core, the issue is that a finite coordinate space induces the need for boundary conditions, which induces the discretization of the momentum-space and the spectrum of the QCD Hamiltonian -- see figure below.

Two-particle scattering amplitude T in a finite and infinite volume. In a finite volume: It is a real-valued and singular function of interaction energy (E: X-axis in units of particle mass M). Position of the poles corresponds to the energy eigenvalues to be measured in a corresponding lattice setup. In the infinite volume: It is holomorphic complex-valued function of energy. Pull the blue bar to switch between both "worlds".

The so-called quantization condition bridges these two worlds (finite and infinite volume interaction spectra). The general form of such a condition is not known. Two- and three-body quantization conditions are however available now. In particular the latter are quite new and the work on those including first applications is currently (2024) the frontier of the research. Alternative methods aiming in extracting bulk properties of infinite-volume quantities from a larger set of finite volume configurations exist as well. For a recent dedicated review of these and other approaches see:
Recently we have organized a dedicated workshop on this research at the Bethe-Center in for Theoretical Physics in Bonn (Germany) - " Multihadron Dynamics in a Box". In recent years, I have worked on the derivation of three-body quantization condition, the corresponding infinite-volume amplitude, and applications to physical three-body systems calculated on the lattice. Some relevant references are listed below:

Unphysical pion mass
a chance and a challenge

In many lattice calculations the pion mass is higher than the physical one for technical reasons. This technicality needs to be overcome (in addition to infinite volume mapping and continuum extrapolations) when comparing lattice results to the physical ones. On the other side, it also is a chance to explore the structure of QCD promoting a pion mass to a continuous parameter, and exploring our understanding in this orthogonal to the energy dimension.

Two-pion scattering amplitude at twice the physical pion mass from Lattice QCD (colored points with error bars) vs. phenomenological model (red line) based on Chiral Perturbation Theory. [source]
Chiral Perturbation Theory (see below) dictates the pion mass dependence of QCD Green's functions allowing for the so-called chiral extrapolations. In this context, I have studied several physical systems with and without resonances:
``The rho-resonance with physical pion mass from Nf=2 lattice QCD'' (2020)
``A cross-channel study of pion scattering from lattice QCD'' (2019)
``Pion scattering in the isospin I=2 channel from elongated lattices'' (2019)
``Extraction of isoscalar ππ phase-shifts from lattice QCD'' (2018)
``Chiral symmetry constraints on resonant amplitudes'' (2017)
``Chiral Extrapolations of the ρ(770) Meson in Nf=2+1 Lattice QCD Simulations'' (2017)
``Chiral Extrapolation of the Sigma Resonance'' (2016)
``Finite volume effects and quark mass dependence of the N(1535) and N(1650)'' (2013)


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