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Dmitry Diakonov, Victor Petrov, and Maxim Polyakov were my colleagues and collaborators. Mitya and Vitya were also, and maybe first of all, my close personal friends for many many years. What follows is a mixture of some recollections and a review of our joint works.

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In this short note I recall my friendship with Mitya Diakonov against the background of the historical and political events of that time.

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Brief personal recollections about Professor Dmitry Diakonov (1949–2012) are presented. Exact solutions to the Schrödinger equation for gluodynamics and quartic tilted anharmonic oscillator are mentioned.

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Collecting my memories below, I decided not to focus on particular physics ideas and their development over time, but to focus on what makes Dimitry Diakonov so special. Using his family background, based on linguistics and history, he developed over the years multiple interests and skills reaching far outside physics.

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Dedication to my untimely departed friends, Dmitry Igorevich Diakonov, Viktor Yur’evich Petrov, and Maxim Vladimirovich Polyakov.

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The instanton vacuum provides an effective description of chiral symmetry breaking by local topological fluctuations of the gauge fields, as observed in lattice QCD simulations. The resulting effective dynamics at momenta below \(1/\bar \rho \approx 0.6\) GeV explains the basic features of light-quark correlation functions and is used extensively in studies of hadron structure. The instanton fields also make definite contributions to the gluonic structure of light hadrons, as expressed in the matrix elements of composite quark–gluon or gluon operators. The article reviews the gluonic structure of light hadrons (nucleon, pion) induced by instantons. This includes: (i) twist-2 parton distributions and momentum sum rule; (ii) twist-3 angular momentum and spin-orbit interactions; (iii) twist-3 and twist-4 quark–gluon correlations and power corrections; (iv) trace anomaly and hadron mass decomposition; (v) scalar gluon form factors and mechanical properties; (vi) axial anomaly and pseudoscalar gluon form factors. It also discusses possible further applications of the methods and recent developments including gauge field configurations beyond instantons.

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\({\mit \Theta }^+\) is a putative light pentaquark state of positive parity with minimal quark content \((uudd\bar {s})\). It naturally emerges in chiral models for baryons, but experimental evidence is uncertain. We review the theoretical foundations of chiral models and their phenomenological applications to exotic states. In particular, we discuss in detail the pentaquark widths with special emphasis on the cancellations occurring in the decay operator. We also discuss some experiments, mainly those whose positive evidence of \({\mit \Theta }^+\) persists to this day. This review is dedicated to Dmitry Diakonov, Victor Petrov, and Maxim Polyakov and their contribution to the \({\mit \Theta }^+\) story.

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In this paper, I pay tribute to my exceptional colleagues and friends Dmitri Diakonov, Victor Petrov, and Maxim Polyakov by examining the experimental progress and current status of the searches of the \({\mit \Theta }^+\) pentaquark from its inception to the present.

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This brief review is dedicated to the memory of Maxim V. Polyakov and his pioneering contributions to pentaquark physics. We focus on his seminal 1997 work with Diakonov and Petrov that predicted the \({\mit \Theta }^+\) pentaquark, a breakthrough that initiated an intense period of research in hadron physics. The field faced a significant setback when the CLAS Collaboration at Jefferson Lab reported null results in 2006, leading to a dramatic decline in light-pentaquark research. Nevertheless, Maxim maintained his scientific conviction, supported by continued positive signals from the DIANA and LEPS collaborations. Through recent experimental findings on the \({\mit \Theta }^+\) and the nucleon-like resonance \(N^*(1685)\), we examine how Polyakov’s theoretical insights, particularly the prediction of a narrow width (\({\mit \Gamma } \approx 0.5\)–\(1.0\) MeV), remain relevant to our understanding of the \({\mit \Theta }^+\) light pentaquark.

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In this short note, I would first like to introduce several conversations that we had with Mitya around the years 2010–2012 when he stayed at RCNP, Osaka. Then a possible interpretation of the parity of the pentaquark is discussed.

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This paper describes my personal appreciation for some of the great research achievements of Mitya Diakonov, Vitya Petrov, and Maxim Polyakov, and how my own research career has followed the paths they opened. Among the topics where they have been the most influential have been the pursuit and study of the exotic pentaquark. The search for exotics may require a complementary approach, such as experimental and theoretical activity. Here, I would like to focus on a story of \(N(1680)\), a non-strange unitary partner of \({\mit \Theta }^+\), in which I was involved.

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We present prospects for the \({\mit \Theta }^+\) pentaquark baryon search using the newly constructed LEPS2 facility at SPring-8. The LEPS2 detector system features significant improvements in acceptance for multi-particle final states compared to previous experiments. Our search employs two complementary strategies: direct production in the \(\gamma n \to K^-{\mit \Theta }^+\) reaction using a liquid deuterium target with a photon beam up to 2.4 GeV, and \(\bar {K}^{*0}\)-associated \({\mit \Theta }^+\) production using a liquid hydrogen target with a photon beam up to 2.9 GeV. The extended acceptance covers both forward and large-angle regions, effectively spanning the kinematic regions explored by previous LEPS and CLAS experiments. The large acceptance and improved resolution of LEPS2, combined with these complementary approaches, provide unprecedented sensitivity for establishing the existence of the \({\mit \Theta }^+\) or placing definitive upper limits on its production.

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In the search for the pentaquark \({\mit \Theta }^+\) baryon, the world data on \(K^+n\to K^+n\) and \(K^+n\to K^0p\) reactions on the deuteron target and \(K^0p\to K^+n\) and \(K^0_{\mathrm {L}}\,p\to K^0_{\mathrm {S}}\,p\) reactions on the hydrogen target are revisited to study the isoscalar component of the scattering amplitudes. The determination of the \(s\)- and \(p\)-wave phase shifts for the two former channels on the deuteron target is presented in the low-momentum region with the uncertainty due to the deuteron form factors at forward angles discussed. For the reactions on hydrogen targets, a similarity expected from time reversal between the charge exchange processes \(K^+n\to K^0p\) and \(K^0p\to K^+n\) is exploited, while we present the analysis of the \(K_{\mathrm {L}}^0\,p\to K_{\mathrm {S}}^0\,p\) scattering based on the \(s\)-wave phase shift for low momenta combined with the \(t\)-channel \(\rho ^0(775)+\omega (782)+\phi (1020)\) vector meson exchange at high momenta. With the mass of 1535 MeV and decay width of 5 MeV, the possibility of the \({\mit \Theta }^+\) in the Breit–Wigner (BW) form is tested in the differential cross section measured by Damerell et al. [Nucl. Phys. B 94, 374 (1075)] at \({P_\mathrm {Lab}}=434\) MeV/\(c\) and polarization. Our analysis suggests measuring the polarization observable rather than the differential cross section, which has sufficient sensitivity to distinguish the \({\mit \Theta }^+\) baryon among all spin and parity states \(J^P = 1/2^\pm \) and \(3/2^\pm \).

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This review deals with measurements and future experiments of light pentaquark searches using hadron beams.

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Dmitri Diakonov has played a significant role in identifying the degrees of freedom underlying hadron spectroscopy. His contributions are discussed with a view on recent developments.

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The interpretation of the energy-momentum tensor form factor \(D(t)\) of hadrons in terms of pressure and shear force distributions is discussed, concerns raised in the literature are reviewed, and ways to reconcile the concerns with the interpretation are indicated.

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We present general features of the transverse densities of the stress-energy-momentum tensor \({\mit \Theta }^{\mu \nu }\) in the pion. We show positivity of the transverse density of \({\mit \Theta }^{++}\) (analogous to the positivity of the transverse density of the electromagnetic current \(J^+\)) and discuss its consequences in conjunction with analyticity and quark–hadron duality, as well as the connection to \(\pi \pi \) scattering at low energies. Our analysis takes into account the perturbative QCD effects, dominating at high momenta (or low transverse coordinate \(b\)), the effects of Chiral Perturbation Theory, dominating at low momenta (high \(b\)), and meson dominance in the intermediate region. We incorporate constraints form analyticity, leading to sum rules for the spectral densities of the corresponding form factors, which ia. are relevant for the high-momentum (or the low-\(b\)) asymptotics. With the obtained high- and low-\(b\) behavior, we deduce that the scalar (trace-anomaly) gravitational transverse density \({\mit \Theta }^{\mu }_\mu (b)\) must change the sign, unlike the case of the positive definite \(J^+(b)\) or \({\mit \Theta }^{++}(b)\). We also discuss the transverse pressure in the pion, which is positive and singular at low \(b\), and negative at high \(b\), in harmony with the stability criterion. The results for the form factors for space-like momenta are compared to the recent lattice QCD data.

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The gravitational form factors of the deuteron are calculated in the framework of non-relativistic chiral effective field theory. Non-relativistic reduction of the matrix element of the energy-momentum tensor operator for spin-one systems is worked out, and the gravitational form factors of the deuteron are extracted from the three-point function of the energy-momentum tensor using the LSZ reduction formula. The obtained form factors are compared to results of model calculations available in the literature.

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One method for deriving a factorization for QCD processes is to use successive integration over fields in the functional integral. In this approach, we separate the fields into two categories: dynamical fields with momenta above a relevant cutoff, and background fields with momenta below the cutoff. The dynamical fields are then integrated out in the background of the low-momentum background fields. This strategy works well at tree level, allowing us to quickly derive QCD factorization formulas at leading order. However, to extend the approach to higher loops, it is necessary to rigorously define the functional integral over dynamical fields in an arbitrary background field. This framework was carefully developed for the calculation of the effective action in a background field at the two-loop level in the classic paper by Abbott «The Background Field Method Beyond One Loop», Nucl. Phys. B 185, 189 (1981). Building on this work, I specify the renormalized background-field Lagrangian and define the notion of the quantum average of an operator in a background field, consistent with the “separation of scales” scheme mentioned earlier. As examples, I discuss the evolution of the twist-2 gluon light-ray operator and the one-loop gluon propagator in a background field near the light cone.

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Motivated by experimental data at large momentum transfer, we update the analysis of the two-photon exchange effect in the electron–nucleon scattering using the effective field theory formalism. Our approach is suitable for describing the hard \(s\sim |t|\sim |u|\gg \mit {\Lambda }^2\) region, where the hadronic model calculations are not accurate enough. We improve the estimates of various long-range matrix elements and discuss the obtained numerical effects for the unpolarised elastic cross section. Assuming a linear behaviour of the reduced cross section with respect to the photon polarisation, we show that the obtained description allows us to resolve the form factor discrepancy for \(Q^2=2.5\)–3.5 GeV\(^2\). However, the effect obtained is quite small for higher values of \(Q^2\). It is possible that nonlinear effects may be important in understanding the discrepancy in this region. Estimates of the elastic electron–neutron cross section in the region of \(Q^2=2.5\)–3.5 GeV\(^2\) are also performed. The obtained TPE effects are sufficiently large and must be taken into account.

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We summarize the reasons for using the light-cone mechanics of nuclei for description of high-energy nuclear processes and describe the steps, which have led to the discovery of high-momentum correlations in nuclei.

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A factorization theorem for the hard coherent processes involving high-energy scattering of photons is derived. We elaborate the treatment for the case of the longitudinally polarized photons and extend the treatment to the case of the processes initiated by transversely polarized photons, as well as to the coherent photo-production of heavy quarkonia. A wave function of a hadron being the solution of the Schrödinger equation is presented in the space-time evolution of initially zero-size and zero-color charge wave packet of bare quarks and gluons. The increase of the running coupling constant in QCD with the distance leads to the coexistence of two phases within the hadron wave function. One phase is the pQCD phase, the second phase is dominated by the phenomenon of spontaneously broken chiral symmetry. The critical line at \(r=r_{\mathrm {c}}\) follows from the gauge symmetry, asymptotic freedom, causality, quantum diffusion, and from the cancellation of soft color fields outside the wave packet. The distinctive properties of QCD are the significant probability of a high-momentum tail of the minimal Fock component (MFC) of the hadron wave function and the strong suppression of the soft part of MFC of hadron WF in the strong coupling regime of QCD. The factorization theorems predict hard coherent processes, color transparency, color fluctuations phenomena, etc. We argue that the value of the radius of the region occupied by the pQCD phase, \(r_{\mathrm {c}}\), is restricted by the radius of the onset of spontaneously broken chiral symmetry \(r_{\mathrm {c}}\gg r_{\mathrm {CS}}\). The success of the constituent quark model for a baryon suggests that the radius of the confinement of color is significantly larger than \(r_{\mathrm {c}}\). The presence of a color core in hadrons matches well with preQCD field theories at distances larger than the critical line for the onset of the regime of spontaneously broken chiral symmetry. It substitutes Landau nullification of the interaction within preQCD field theories, the preQCD Feynman model at the distances smaller than \(r_{\mathrm {c}}\).

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We consider properties of the inhomogeneous solution found recently for the \(\mathbb {CP}^{\,N-1}\) model. The solution was interpreted as a soliton. We reevaluate its energy in three different ways and find that it is negative contrary to the previous claims. Hence, instead of the solitonic interpretation, it calls for reconsideration of the issue of the true ground state. While complete resolution is still absent, we show that the energy density of the periodic elliptic solution is lower than the energy density of the homogeneous ground state. We also discuss similar solutions for the \({\mathbb {O}}(N)\) model and for SUSY extensions.