Gribov-85 Memorial Volume: Exploring Quantum Field Theory

The following sections are included:

• PREFACE
• CONTENTS

I attended several of Gribov's legendary seminars. He seemed permanently inspired. It was a deep pleasure to follow him in his arguments…

It was always a pleasure to remember your husband Gribov, and how much it meant to spend time with him. Although I am not a physicist, it is clear what a great scientist Volodya was, innovative and deep. At the same time he was an extremely nice person.

He was not only a great scientist, but a person who, apart from his great contributions to physics, was also able to clearly see what was right and what was wrong in life. His presence, his opinion about people and science was always a delightful experience. Albert was the only other person I have known well who also clearly knew what was right and what was wrong, and was willing to stand up for his principles. We are lucky to have known and lived with such great individuals.

Note from Publisher: This article contains the abstract only.

V.N. Gribov was incapable of sparing himself. All his life he has acted as passionately, as intensively as he did in his youth when he worked with Landau and Pomeranchuk and led the Theoretical Physics Department of the Ioffe Institute and later of the Nuclear Physics Institute in Leningrad. For him “leading” meant just hiring the most talented students to join the institute and then engaging in merciless, endless but fruitful discussions with them on the problems they worked on. He did not care about who was right, he cared only about the right answer. Incidentally, only in physics did he push tirelessly to reach a decision. In conversations about literature or politics as a rule he argued mildly and kindly. Not that he did not have thoughtful and firm opinions about many questions, but because he seemed to realize that in these matters different points of view were arguable…

The talk presents the main lines of biography of the prominent physicist and bright personality. Also given is a necessarily brief description of Gribov's scientific work.

The close friendship between Volodya Gribov and myself dates back to the 1970 Rochester High Energy Conference, which took place in Kiev. At that time I was a young Visiting Associate Professor in Berkeley, CA, active, like most of G.F. Chew group members, in both theory and phenomenology studies of S-Matrix Theory of Regge Poles. The Berkeley school approach was initiated by Chew and Frautschi in their 1961 paper, which led to the bootstrap theory. Less known in the West were Gribov's late 1950s and early 1960s papers which led to the definition of the Pomeron, the Reggeon field theory and the Pomeron partonic decomposition. In the early 1970 I was invited to give a talk at the Kiev Rochester Conference. I was very naive thinking that being invited would get me a visa to the Soviet Union, even though it had broken its diplomatic relations with Israel. A short visit to the Soviet Consulate in San Francisco ended with a rude and very clear statement that I did not have a chance to get the desired visa. Luckily, quite a few senior colleagues, who were also invited, convinced the Soviet authorities to grant a small number of Israelis, myself included, the desired visa, provided that we apply in Europe. I had to fly from San Francisco to Helsinki to get my visa and proceed via Leningrad and Moscow on my way to Kiev. Arriving in Kiev I was confronted by a very friendly person, introducing himself as Volodya Gribov. I knew that he was one of the Soviet stars of particle physics, but not much more. Volodya suggested that we should spend our free time in Kiev parks, where we could talk freely. Volodya's suggestion was too exotic to be declined, and we ended spending lots of hours, night after night, in Kiev parks, changing our route each night. Most of the time we were alone, joined by Lev Okun a couple of times. Our conversations centred, at the beginning, on Israeli politics, the Palestinian conflict with Israel and other obvious subjects. The issue of massive immigration from the Soviet Union to Israel following the collapse of the Soviet Union seemed too exotic to even be discussed. Volodya kept asking me about the quality of physics teaching and research back home, but, obviously, this subject was secondary on his priority list. What surprised me was how quickly we started to discuss also very personal issues. In general, I refrain from personal discussions on very private matters. But there was something touching in Volodya's wish to talk. A week later, when the Kiev Conference ended, both of us realised that we have become very close friends. I met Volodya again a year after his son Lenya died in a mountaineering accident, shortly after completing his Ph.D. It was a very sad meeting and in my opinion Volodya partially recovered thanks to Júlia. Years later, I still miss him…

It may sound too trivial, but it is very important that life consists of details. It becomes still more important when you are trying to get together the recollections using the details you remember. Snapshots render memorable moments in imperfect images. This is exactly what is presented below viewed by an external observer…

Time puts things in their places, and as time passes, Volodya's role as a universal physicist with an exceptional intuition becomes more and more significant.

Note from Publisher: This article contains the abstract only.

The following sections are included:

• Bus seminar
• Working with Gribov
• Lenya

I would like to write a few words, in memory of Volodya, for the occasion of this workshop which is organized under his name and will start by publicly remembering his essential presence in our theoretical physics world. We had in Orsay (LPT, theoretical physics laboratory) the incredible luck to meet him in 1992 for the first time when we could finally invite this famous and celebrated figure of modern particle physics. He gave a series of lectures this year: Orsay lectures on confinement–in which he mainly developed the picture of confinement based on light quarks that led to many discussions which contributed to open the road of the search actually still going on…The way he was giving his talks nobody will forget. He always started by describing the field he would talk about in a very passionate and extraordinary way: he would say: “I have a picture” which would force even the most far away spectators to participate in an actively engaged vision of the problem he was talking about. He was building with enthusiasm the theoretical image which led to the result he wanted to show. He will remain in our memory as a rare model of intellectual passion. This led him to formulate in a unique way precious theoretical results. Thank you Volodya…

Note from Publisher: This article contains the abstract only.

The current status of ultrahigh energy diffractive collisions is given. Profile function and K-matrix function techniques in the impact parameter space are presented, and the Gribov universality of the hadron total cross sections is discussed.

The theory of the high energy scattering in QCD and gravity is based on the reggeization of gluons and gravitons, respectively. We discuss the corresponding effective actions for reggeized particle interactions. The Euler-Lagrange equations in these theories are constructed with a variational approach for the effective actions and by using their invariance under the gauge and general coordinate transformations.

High energy scattering processes in QCD can be described in terms of interacting pomerons. Vertices for the interaction of pomerons have been derived in perturbative QCD. Their properties hint at the existence of a 2+1-dimensional conformal field theory of interacting pomerons. Here we study in particular the 1-to-2 and the 1-to-3 pomeron vertex and find them to be related to each other by a conformal bootstrap relation.

Pomeron and non vacuum reggeon parameters which reasonably describe total and elastic cross sections are obtained. The ratio of the real and imaginary part of the forward elastic scattering amplitude is calculated with these parameters and it coincides with the experimental data in the whole energy region. The shower enhancement coefficient in the quasi-eikonal approach appears to be less than unity which corresponds to the definite distribution of parton showers in incident hadrons.

Reggeon unitarity and non-Abelian gauge field copies are focused on as two Gribov discoveries that, it is suggested, may ultimately be seen as the most significant and that could, in the far distant future, form the cornerstones of his legacy. The crucial role played by the Gribov ambiguity in the construction of gauge theory bound-state amplitudes via reggeon unitarity is described. It is suggested that the existence of a physical, unitary, S-Matrix in a gauge theory is a major requirement that could even determine the theory.

We explain that coherence of high energy QED and QCD processes implies existence of new kind of phenomena which are beyond a framework based on Regge poles (cuts). New phenomena emerge as the consequence of compositeness of the bound states and the Lorentz slowing down of interaction. We focus on the color fluctuations phenomena predicted earlier for pA collisions within QCD and recent evidence for this phenomenon from pA LHC run, significant modification of nuclear shadowing phenomenon in the diffractive photoproduction of vector mesons observed recently in the ultra peripheral collisions at LHC. We outlined briefly general properties of color fluctuations phenomena and perspectives of future studies of this phenomenon in electron (photon) collisions with nuclei.

In high energy heavy ion collisions of RHIC and LHC, a strongly interacting quark gluon plasma (sQGP) is created. This medium undergoes a hydrodynamic evolution, before it freezes out to form a hadronic matter. The initial state of the sQGP is determined by the initial distribution of the participating nucleons and their interactions. Due to the finite number of nucleons, the initial distribution fluctuates on an event-by-event basis. The transverse plane anisotropy of the initial state can be translated into a series of anisotropy coefficients or eccentricities: second, third, fourth-order anisotropy etc. These anisotropies then evolve in time, and result in measurable momentum-space anisotropies, to be measured with respect to their respective symmetry planes. In this paper we investigate the time evolution of the anisotropies. With a numerical hydrodynamic code, we analyze how the speed of sound and viscosity influence this evolution.

A new exact and analytic solution of non-relativistic fireball hydrodynamics is presented. It describes an expanding three-axis ellipsoid that rotates along one of its principal axes. The observables are calculated using simple analytic formulas. Following earlier works, it is pointed out that azimuthal oscillation of the off-diagonal Bertsch-Pratt radii of Bose-Einstein correlations as well as rapidity dependent directed and third flow measurements provide means to determine the magnitude of the rotation of the fireball. It is argued that observing this rotation and its dependence on collision energy may lead to new information on the equation of state of the strongly interacting quark gluon plasma produced in high energy heavy ion collisions.

I begin with a tribute to V.N. Gribov and then come to a particular problem which would be of interest for him. His first paper on reggeology was devoted to elastic scatterings of hadrons. Here, using the unitarity relation in combination with experimental data about the elastic scattering in the diffraction cone, I show how the shape and the darkness of the interaction region of colliding protons change with the increase of their energies. In particular, the collisions become fully absorptive at small impact parameters at LHC energies that results in some special features of inelastic processes as well. The possible evolution with increasing energy of the shape from the dark core at the LHC to the fully transparent one at higher energies is discussed. It implies that the terminology of the black disk would be replaced by the black torus.

Shadow effects at collisions of hadrons with light nuclei at high energies were subject of scientific interest of V.N. Gribov, first, we mean his study of the hadron-deuteron scattering, see Sov. Phys. JETP 29, 483 (1969) [Zh. Eksp. Teor. Fiz. 56, 892 (1969)] and discovery of the reinforcement of shadowing due to inelastic diffractive rescatterings. It turns out that the similar effect exists on hadron level though at ultrahigh energies…

Diffractive production is considered in the ultrahigh energy region where pomeron exchange amplitudes are transformed into black disk ones due to rescattering corrections. The corresponding corrections in hadron reactions h₁ + h₃ → h₁ + h₂ + h₃ with small momenta transferred (, ) are calculated in terms of the K-matrix technique modified for ultrahigh energies. Small values of the momenta transferred are crucial for introducing equations for amplitudes. The three-body equation for hadron diffractive production reaction h₁ + h₃ → h₁ + h₂ + h₃ is written and solved precisely in the eikonal approach. In the black disk regime final state scattering processes do not change the shapes of amplitudes principally but dump amplitudes by a factor ~ ¼; initial state rescatterings result in additional factor ~ ½. In the resonant disk regime initial and final state scatterings damp strongly the production amplitude that corresponds to at in this mode.

Gribov's partonic Pomeron provides the foundations of updated models which incorporate soft and hard scattering, so as to reproduce the recent LHC p-p cross sections. Explicitly, total, elastic, inelastic and diffrative data. Leading models are: GLM (Gotsman, Levin, Maor), KMR (Khoze, Martin, Ryskin), Kaidalov-Poghosyan and Ostapchenco.

None of these models in their pre-LHC versions reproduced the TOTEM, ALICE, ATLAS and CMS soft LHC data, needing considerable reconstructions, either in the fitting procedures (GLM), or in the details of the theoretical models. In the following, I shall relate mostly to the GLM model, emphasizing the critical role of the diffractive channels.

The dipole phenomenology, which has been quite successfully applied to various hard reactions, especially on nuclear targets, is applied for calculation of Gribov inelastic shadowing. This approach does not include ad hoc procedures, which are unavoidable in calculations done in hadronic representation. Several examples of Gribov corrections evaluated within the dipole description are presented.

Following earlier findings, we argue that the low-|t| structure in the elastic diffractive cone, recently reported by the TOTEM Collaboration at 8 TeV, is a consequence of the threshold singularity required by t-channel unitarity, such as revealed earlier at the ISR. By using simple Regge-pole models, we analyze the available data on the pp elastic differential cross section in a wide range of c.m. energies, namely those from ISR to LHC8, obtaining good fits of all datasets. This study hints at the fact that the non-exponential behavior observed at LHC8 is a recurrence of the low-|t| “break” phenomenon, observed in the seventies at ISR, being induced by the presence of a two-pion loop singularity in the Pomeron trajectory.

QCD possesses a compact gauge group, and this implies a non-trivial topological structure of the vacuum. In this contribution to the Gribov-85 Memorial volume, we first discuss the origin of Gribov copies and their interpretation in terms of fluctuating topology in the QCD vacuum. We then describe the recent work with E. Levin that links the confinement of gluons and color screening to the fluctuating topology, and discuss implications for spin physics, high energy scattering, and the physics of quark-gluon plasma.

A new picture of quark confinement based on the instability of Coulomb phase at low energy was introduced by Volodya Gribov in the early nineties. In QCD the effective α coupling constant can reach very large values in the infrared regime what generates Coulomb phase instabilities. In the Gribov picture the instability leads to a vacuum decay into light quarks for coupling constants α larger than a critical value , for SU(N) gauge theories. The instability of Coulomb phase can be derived from first principles in any non-Abelian gauge theory for , a value which is larger than the Gribov critical value. In this paper we review the analytic derivation of the Gribov mechanism from first principles and analyze the effects of dynamical quarks in the instability of the Coulomb phase. The instabilities associated to light quarks turn out to appear at larger values of α than the ones induced from pure gluon dynamics, unlike it is expected in the standard Gribov scenario. The analytic results confirm the consistency of the picture where quark confinement is mainly driven by gluonic fluctuations.

The apparent nonlocality of the Coulomb gauge external field problem in electrodynamics is illustrated with an example in which nonlocality is especially striking. Explanation of this apparent nonlocal behaviour based on a purely local picture is given. A gauge invariant decomposition of the Lorentz-force into two terms with clear physical meanings is pointed out. Based on this decomposition derivation of the Aharonov–Bohm effect in terms of field strengths alone is given.

The trace anomaly or, equivalently, the interaction measure is an important thermodynamic quantity/observable, since it is very sensitive to the nonperturbative effects in the gluon plasma. It has been calculated and its analytic and asymptotic properties have been investigated with the combined force of analytic and lattice approaches to the SU(3) Yang-Mills (YM) quantum gauge theory at finite temperature. The first one is based on the effective potential approach for composite operators properly generalized to finite temperature. This makes it possible to introduce into this formalism a dependence on the mass gap Δ², which is responsible for the large-scale dynamical structure of the QCD ground state. The gluon plasma pressure as a function of the mass gap adjusted by this approach to the corresponding lattice data is shown to be a continuously growing function of temperature T in the whole temperature range [0,∞) with the correct Stefan-Boltzmann limit at very high temperature. The corresponding trace anomaly has a finite jump discontinuity at some characteristic temperature T_c = 266.5 MeV with latent heat ∈_LH = 1.41. This is a firm evidence of the first-order phase transition in SU(3) pure gluon plasma. It is exponentially suppressed below T_c and has a complicated and rather different dependence on the mass gap and temperature across T_c. In the very high temperature limit its non-perturbative part has a power-type fall off.

We have explicitly shown that Quantum Chromodynamics is a color gauge invariant theory with non-zero mass gap, which has been defined as the value of the regularized full gluon self-energy at a finite scale point. The mass gap itself is mainly generated by the nonlinear interaction of massless gluon modes. All this allows one to establish the structure of the full gluon propagator in the explicit presence of the mass gap. In this case, the two independent general types of formal solutions for the full gluon propagator as a function of the regularized mass gap have been found: (i) The nonlinear iteration solution at which the gluons remain massless is explicitly present. (ii) Existence of the solution with an effective gluon mass is also demonstrated.

Discussion of the physical realization of coordinates demonstrates that the quantum theory of gravity (still absent) should be non-local and, probably, non-commutative as well.

Astronomical observations of recent years show that the universe at high redshifts (about ten) is densely populated by early formed objects: bright galaxies, quasars, gamma-bursters, and it contains a lot of metals and dust. Such a rich variety of early formed objects have not been expected in the standard model of formation of astrophysical objects. There is serious tension between the standard theory and the observations. We describe the model which relaxes this tension and nicely fits the data. The model naturally leads to the creation of cosmologically significant antimatter which may be abundant even in the Galaxy. Phenomenological consequences of our scenario and the possibility of distant registration of antimatter are discussed.

An influence of the rotation and gravity of the Earth on the particle motion and the spin evolution is not negligible and it should be taken into account in spin physics experiments. The Earth rotation brings the Coriolis and centrifugal forces in the lab frame and also manifests in the additional rotation of the spin and in the change of the Maxwell electrodynamics. The change of the Maxwell electrodynamics due to the Earth gravity is much smaller and can be neglected. One of manifestations of the Earth rotation is the Sagnac effect. The electric and magnetic fields acting on the spin in the Earth's rotating frame coincide with the corresponding fields determined in the inertial frame instantly accompanying a lab. The effective electric field governing the particle motion differs from the electric field in the instantly accompanying frame. Nevertheless, the difference between the conventional Lorentz force and the actual force in the Earth's rotating frame vanishes on average in accelerators and storage rings due to the beam rotation. The Earth gravity manifests in additional forces acting on particles/nuclei and in additional torques acting on the spin. The additional forces are the Newton-like force and the reaction force provided by a focusing system. The additional torques are caused by the corresponding focusing field and by the geodetic effect. As a result, the Earth gravity leads to the additional spin rotation about the radial axis which may not be negligible in EDM experiments.

Geometrization of the fundamental interactions has been extensively studied during the century. The idea of introducing compactified spatial dimensions originated by Kaluza and Klein. Following their approach, several model were built representing quantum numbers (e.g. charges) as compactified space-time dimensions. Such geometrized theoretical descriptions of the fundamental interactions might lead us to get closer to the unification of the principle theories. Here, we apply a 3 + 1_C + 1 dimensional theory, which contains one extra compactified spatial dimension 1_C in connection with the flavor quantum number in Quantum Chromodynamics. Within our model the size of the 1_C dimension is proportional to the inverse mass-difference of the first low-mass baryon states. We used this phenomena to apply in a compact star model — a natural laboratory for testing the theory of strong interaction and the gravitational theory in parallel. Our aim is to test the modification of the measurable macroscopical parameters of a compact Kaluza — Klein star by varying the size of the compactified extra dimension. Since larger the R_C the smaller the mass difference between the first spokes of the Kaluza — Klein ladder resulting smaller-mass stars. Using the Tolman — Oppenheimer — Volkov equation, we investigate the M-R diagram and the dependence of the maximum mass of compact stars. Besides testing the validity of our model we compare our results to the existing observational data of pulsar properties for constraints.

Gravitational form factors are the matrix elements of the Belinfante energy momentum tensor (EMT) which naturally incorporate the hadron structure and the equivalence principle. The relocalization property allowing to transform EMT to the Belinfante form provides the “kinematical” counterpart of the famous U_A(1) problem. The equivalence principle may be approximately valid for quarks and gluons separately in non-perturbative (NP)QCD, and this conjecture is supported by the experimental and lattice data. The extradimensional gravity leading to holographic AdS/QCD is supporting the relation of quark transverse momentum to the Regge slope, discovered by V.N. Gribov.

I define a naturalness criterion formalizing the intuitive notion of naturalness discussed in the literature. After that, using ϕ⁴ as an example, I demonstrate that a theory may be natural in the MS-scheme and, at the same time, unnatural in the Gell-Mann–Low scheme. Finally, I discuss the prospects of using a version of the Gell-Mann–Low scheme in the Standard Model.

Calculation of hard three-loop corrections of order mα⁷ to hyperfine splitting in muonium and positronium is reviewed. All these contributions are generated by the graphs with photon, electron and/or muon loop radiative insertions in the two-photon exchange diagrams. We calculate contributions of six gauge invariant sets of diagrams.

We consider the electromagnetic production of positrons and electrons in collisions of slow heavy nuclei. This process is dominated by emission of positrons, with the electrons captured by nucleus.

The production of different quarkonium states provides unique insight to the hot and cold nuclear matter effects in the strongly interacting medium that is formed in high energy heavy ion collisions. While LHC explores the energy frontier, RHIC has a broad physics program to explore the nuclear modification at different energies in a wide range of systems. Some of the most interesting recent results on J/ψ and γ production in p+p, d+Au and A+A collisions from PHENIX and STAR are summarized in this work.

We overview the history of superluminal neutrinos, how they were first observed then refuted, and the emerged physical interpretations. It is shown that in certain cases such false experimental discoveries can help the development of science.

We use the connection between the Frobenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search for a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensures the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has an explicit and simple form in the flat coordinates on the Frobenius manifold in the general case of (p,q) Minimal Liouville gravity.

We study Quantum Electrodynamics in d = 3 (QED3) coupled to N_f flavors of fermions. The theory flows to an IR fixed point for N_f larger than some critical number . For , chiral symmetry breaking is believed to take place. In analogy with the Wilson-Fisher description of the critical O(N) models in d = 3, we use the existence of a fixed point in d = 4 − 2∈ to study the three-dimensional theory. We show how the ∈-expansion can be used to study the anomalous dimension of 2- and 4-fermion operators. The latter leads to an estimate of the critical number . An important novelty compared to the O(N) models is that, because of the structure of spinors, the theory in d = 3 has an enhanced symmetry. We identify the operators in d = 4 − 2∈ that correspond to the additional conserved currents at d = 3.

We study two-dimensional sigma models where the chiral deformation diminished the original 𝒩 =(2, 2) supersymmetry to the chiral one, 𝒩 =(0, 2). Such heterotic models were discovered previously on the world sheet of non-Abelian stringy solitons supported by certain four-dimensional 𝒩 = 1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the β functions in terms of the anomalous dimensions analogous to the NSVZ β function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation.

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field theory at fixed momentum for a spin 1/2 particle along with its antiparticle. It is shown that the essential part of the corresponding automorphism group can naturally be related to the conformal Lorentz group. In addition, the nonsemisimple part of the automorphism group can be understood as “dressing” of the pure one-particle states. The studied mathematical structure may help in constructing quantum field theories in a non-perturbative manner. In addition, it provides a simple example of circumventing Coleman–Mandula theorem using non-semisimple groups, without SUSY.

The scalar mass is determined in the simplest cut-off regularized Yukawa-model in the whole range of stability of the scalar potential. Two versions of the Functional Renormalisation Group (FRG) equations are solved in the Local Potential Approximation (LPA), where also the possible existence of a composite fermionic background is taken into account. The close agreement of the results with previous studies taking into account exclusively the effect of the scalar condensate, supports a rather small systematic truncation error of FRG due to the omission of higher dimensional operators.

Quark–anti quark and diquark condensation have been investigated in the framework of the NJL model. The Pauli–Villars regularization scheme have been used for the investigation of meson condensation in three dimension whereas the four dimensional case has been studied using the Schwinger–Dyson equation considering the Hartree approximation. Diquark condensation in three and four dimension have also been studied considering the Pauli–Villars regularization scheme. Using the Fermi momentum (p_f) of the particle as cutoff parameter, the gap energy/coherence length for meson condensation have been investigated whereas for diquark qq the critical gap energy/critical coherence length (the distance over which there would be no diquark condensation) have been extracted. The variation of the coherence length/gap energy with pf have also been investigated. The results are compared with exciting data. Some interesting observations are made.

We study lattice constructions of gapped fermionic phases of matter. We show that the construction of fermionic Symmetry Protected Topological orders by Gu and Wen has a hidden dependence on a discrete spin structure on the Euclidean space-time. The spin structure is needed to resolve ambiguities which are otherwise present. An identical ambiguity is shown to arise in the fermionic analog of the string-net construction of 2D topological orders. We argue that the need for a spin structure is a general feature of lattice models with local fermionic degrees of freedom and is a lattice analog of the spinstatistics relation.

We present analytic solution of the Dokshitzer.Gribov.Lipatov.Altarelli. Parisi (DGLAP) equation at leading order (LO) in the ϕ³ theory in 6 spacetime dimensions. If the ϕ³ model was the theory of strong interactions, the obtained solution would describe the distribution of partons in a jet. We point out that the local parton-hadron duality (LPHD) conjecture does not work in this hypothetical situation. That is, treatment of hadronisation of shower partons is essential for the description of hadron distributions in jets stemming from proton-proton (pp) collisions at TeV and from electron-positron (e⁺ e⁻) annihilations at various collision energies. We use a statistical model for the description of hadronisation.

Twisted non-Abelian flux-tube solutions are considered in the bosonic sector of a 4-dimensional 𝒩 = 2 super-symmetric gauge theory with U(2)_local × SU(2)_global symmetry, with two scalar doublets in the fundamental representation. Twist refers to a time-dependent matrix phase between the two doublets, and twisted strings have nonzero (global) charge, momentum, and in some cases even angular momentum per unit length. The planar cross section of a twisted string corresponds to a rotationally symmetric, charged non-Abelian vortex, satisfying 1st order Bogomolny-type equations and Gauss-type constraints. Quite unexpectedly some twisted strings lack cylindrical symmetry in ℝ³.

The following sections is included:

• LIST OF PARTICIPANTS AND CONTRIBUTORS