arXiv subjects

Mikhail Shifman

Publications and source records attributed to Mikhail Shifman.

At least 19 recordsLinked to original sources

arXiv2026

More on Classical Stability of Hopf-like Solitons of the Toroidal-Twisted type

The Faddeev-Hopf model [1] supporting Hopfions was shown to emerge in the low-energy limit of four-dimensional scalar quantum electrodynamics (QED) with two charged scalar fields [2, 3]. Faddeev and Noemi conjectured that the Hopfions and Hopf-like solitons -- vortons -- can be based on a twisted toroidal structure inherent to QED [4-6]. This conjecture was discussed in detail in [2] in the approximation of negligibly small extrinsic curvature. Qualitative and semi-quantitative arguments were used to demonstrate the validity of the Faddeev-Noemi hypothesis. Here we further enhance the proof by applying a numerical analysis which confirms that large-size Hopf-like solitons exist as local energy minima in the full QED theory (in the Faddeev-Skyrme model they become topological solitons representing the global minima in the given topological sector).

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arXiv2026

Degenerate vortices and world-line instantons in three-dimensional gauge theories

In this paper we continue the study of particle-like topological solitons with degenerate masses and their mixing due to world-line instantons. Previously, this phenomenon was studied in 1+1-dimensional setups. Here we take a step further and consider degenerate vortices in 2+1 dimensions. We find that, while classically such vortices may be degenerate, they generally mix and split at the quantum level. Supersymmetry protects BPS-saturated vortices only when the number of supercharges in the bulk is large enough.

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arXiv2025

Degenerate kinks and kink-instantons in two-dimensional scalar field models with $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetry

Models with classically degenerate vacua often support quasiclassical configurations of nontrivial topology. In (0+1)-dimensional quantum mechanics with a double-well potential, for example, instantons induce mixing between the two perturbative ground states in the purely bosonic case, while in the supersymmetric version, the tunneling amplitude is suppressed. In this work, we investigate (1+1)-dimensional models featuring classically Bogomol'nyi-Prasad-Sommerfield saturated kinks with degenerate masses and identical topology. Recent studies suggest that such kinks may undergo mixing mediated by scalar-field instantons. We analyze this phenomenon in a supersymmetric framework and demonstrate that, whereas mixing indeed occurs in the bosonic theory, the presence of fermionic zero modes in the supersymmetric case leads to the vanishing of the transition amplitude. To illustrate these results, we examine two examples featuring Wess-Zumino models with two and four supercharges. The latter example is motivated by the Affleck-Dine-Seiberg superpotential. We also present a number of developments of instanton calculus in the case of instantons in kink backgrounds.

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arXiv2025

Two types of domain walls in $\mathcal{N}=1$ super-QCD: how they are classified and counted

We study multiplicities and junctions of BPS domain walls interpolating between different chiral vacua in $\mathcal{N}=1$ supersymmetric QCD (SQCD) with the SU$(N)$ gauge group and a varying number of fundamental quarks. Depending on the number of flavors $F$, two distinct classes of {\em degenerate} domain walls emerge: (i) locally distinguishable, i.e., those which differ from each other locally, in local experiments; and (ii) those which have identical local structure and are differentiated only topologically, through judiciously chosen compactifications. In the first class, two-wall junctions exist, while in the second class, such junctions do not exist. Acharya and Vafa counted {\em topologically distinguishable} walls in pure super-Yang-Mills. Ritz, Shifman, and Vainshtein counted the {\em locally distinguishable} walls in $F=N$ SQCD. In both cases, the multiplicity of $k$ walls was the same, $ν_{N,k}^\text{walls}= N!/\big[(N-k)!k!\big]$. We study the general case $0\leqslant F\leqslant N$, with mixed sets of walls from both classes (i) and (ii) simultaneously, and demonstrate that the above overall multiplicity remains intact. We argue that the growth of the quark masses exhibits no phase transition at any finite mass. The locally distinguishable walls can turn into topologically distinguishable ones only at $m=\infty$. The evolution of the low-energy wall worldsheet theory in the passage from small to large $m$ is briefly discussed. We also propose a candidate for the low-energy description of wall junctions. The tools used are localization of instantons, supersymmetry enhancement on the walls, and circle compactification.

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arXiv2025

Degenerate Domain Walls in Supersymmetric Theories

In supersymmetric Yang-Mills theories (SYM) tension-degenerate domain walls are typical. Adding matter fields in fundamental representation we arrive at supersymmetric QCD (SQCD) supporting similar walls. We demonstrate that the degenerate domain walls can belong to one of two classes: (i) locally distinguishable, i.e. those which differ from each other locally (which could be detected in local measurements); and (ii) those which have identical local structure and are differentiated only topologically, through a judicially chosen compactification of $\mathbb{R}^4$. Depending on the number of flavors $F$ and the pattern of Higgsing both classes can coexists among SQCD $k$ walls interpolating between the vacua $n$ and $n+k$. We prove that the overall multiplicity of the domain walls obtained after accounting for both classes is $ν_{N,k}^\text{walls}= N!/\big[(N-k)!k!\big]$, as was discovered previously in limiting cases. (Here $N$ is the number of colors.) Thus, $ν_{N,k}^\text{walls}$ is a peculiar index. For the locally distinguishable degenerate domain walls we observe two-wall junctions, a phenomenon specific for supersymmetry with central extensions. This phenomenon does not exist for topological replicas.

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arXiv2024

New Soft Theorems for Two-Scalar Sigma Models

In this paper, we study the scattering amplitudes and soft theorems for the sigma models with two scalars. We show that if the particles are Goldstone bosons, then you necessarily get Adler zero with no possibility for non-trivial soft theorems. For non-Goldstone bosons, the soft behavior is generically captured by the geometric soft theorem studied by Cheung et al., and the right-hand side contains derivatives of lower-point amplitudes. Inspired by the recent work on the 2D sigma models, we study one special two-scalar sigma model, where the presence of symmetries in the target space translates into a special but non-trivial soft theorem without derivatives. We further generalize the construction to two larger classes of such models and derive certain soft theorem sum rules, again avoiding the derivatives of amplitudes. Our analysis provides an interesting hierarchy of two-scalar sigma models and soft theorems, ranging from Goldstone boson case to a generic target space, and showing that there are interesting theories in between.

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arXiv2024

Lie-algebraic Kähler sigma models with the U(1) isotropy

We discuss various questions which emerge in connection with the Lie-algebraic deformation of $\mathbb{CP}^1$ sigma model in two dimensions. First we supersymmetrize the original model endowing it with the minimal ${\cal N}=(0,2)$ and extended ${\cal N}=(2,2)$ supersymmetries. Then we derive the general hypercurrent anomaly in the both cases. In the latter case this anomaly is one-loop but is somewhat different from the standard expressions one can find in the literature because the target manifold is non-symmetric. We also show how to introduce the twisted masses and the $θ$ term, and study the BPS equation for instantons, in particular the value of the topological charge. Then we demonstrate that the second loop in the $β$ function of the non-supersymmetric Lie-algebraic sigma model is due to an infrared effect. To this end we use a supersymmetric regularization. We also conjecture that the above statement is valid for higher loops too, similar to the parallel phenomenon in four-dimensional ${\cal N}=1$ super-Yang-Mills. In the second part of the paper we develop a special dimensional reduction -- namely, starting from the two-dimensional Lie-algebraic model we arrive at a quasi-exactly solvable quantum-mechanical problem of the Lamé type.

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arXiv2023

First-order formalism for $β$ functions in bosonic sigma models from supersymmetry breaking

We consider the renormalization group flow equation for the two-dimensional sigma models with the Kähler target space. The first-order formulation allows us to treat perturbations in these models as current-current deformations. We demonstrate, however, that the conventional first-order formalism misses certain anomalies in the measure, and should be amended. We reconcile beta functions obtained within the conformal perturbation theory for the current-current deformations with traditional ``geometric" results obtained in the background field methods, in this way resolving the peculiarities pointed out in [JHEP10(2023)097]. The result is achieved by the supersymmetric completion of the first-order sigma model.

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arXiv2023

Relating $β'_$ and $γ'_{Q}$ in the ${\cal N}=1$ SQCD Conformal Window

In this note we show that $β'_*$, the $β$-function slopes in the electric and magnetic theories are equal at the corresponding infrared fixed points. This follows from the scaling of the correlators of the trace of the energy momentum tensors. The slopes $β'_*$ determine the scaling dimensions. Our paper can be considered as a commentary to Anselmi et al. [1] -- it proposes an improved derivation not based on a rather contrived construction by Kutasov et al. [2]. As a byproduct we note that $γ'_{Q^*}$ -- the slopes of the matter superfield anomalous dimension -- vanish at both edges of the conformal window where one of the dual theories is strongly coupled. Finally, we determine the two-coupling magnetic fixed point at weak coupling correcting the result of [3]}.

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arXiv2023

Peculiarities of beta functions in sigma models

In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first-order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to $β$ functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. Moreover, in certain cases the first-order calculations produce results that are not symmetric under the classical diffeomorphisms of the target space. Although we could not present the full solution to this remarkable phenomenon, we found some indirect arguments indicating that an anomaly similar to that established in supersymmetric Yang-Mills theory might manifest itself starting from the second loop. We discuss why the difference between two answers might be an infrared effect, similar to that in $β$ functions in supersymmetric Yang-Mills theories. In addition to the generic Kähler target spaces we discuss in detail the so-called Lie-algebraic sigma models. In particular, this is the case when the perturbed field $G^{i\bar j}$ is a product of the holomorphic and antiholomorphic currents satisfying two-dimensional current algebra.

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arXiv2023

Remarks on Baby Skyrmion Lie-Algebraic Generalization

We discuss generalized baby Skyrmions emerging in a (1+2)-dimensional $σ$ model with a certain Lie-algebraic structure. The same result applies to the Polyakov-Belavin instantons in $D=1+1$. The O(3) symmetry of the target space is lost, but O(2) is preserved in the simplest model under consideration. Both the topological charge and the soliton mass (the instanton action) are determined. Of special interest are limiting cases of the deformation parameter $k$ (also referred to as the elongation parameter). If $|k-1|\ll 1$, we arrive at a model which is studied in condensed matter. If $k\gg1$, we obtain the so-called cigar, or sausage model, well-known in little string theory. If $k=1$, we return to CP(1). The deformed model under consideration interpolates between CP(1) and the cigar model. In $D=2$ we calculate the coupling constants renormalization at one loop. At $k\geq 1$ this class of models is asymptotically free in the ultraviolet limit and enters the strong coupling domain in the infrared. Also, in the infrared $k\to 1$ (i.e., we recover CP(1)).

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arXiv2023

$U(1)$ Defects on Domain Lines

Based on recent experimental results, we give field-theoretic description of $U(1)$ defects localized on the domain lines on thin films. We describe topology of our model and solve this model in the adiabatic approximation. It turns out that such a model naturally provides periodic structure observed in experiment. The effective theory turns out to be the sine-Gordon model, but unlike the previous theoretical considerations we argue that in this case it is favorable for sine-Gordon kinks to merge into one defect with a uniform winding. We consider a system of adjacent domain lines and anti-lines and explain the experimental fact that the appearance of defects on a domain line prevents defect creation on the adjacent anti-lines. We also quantize the model and investigate possible effects of finite transverse dimension of the film.

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arXiv2023

Spectral Flow in Instanton Computations and the \boldmath{$\b$} functions

We discuss various differences in the instanton-based calculations of the $β$ functions in theories such as Yang-Mills and $\mathbb{CP}(N\!-\!1)$ on one hand, and $λϕ^4$ theory with Symanzik's sign-reversed prescription for the coupling constant $λ$ on the other hand. Although the aforementioned theories are asymptotically free, in the first two theories, instantons are topological, whereas the Fubini-Lipatov instanton in the third theory is topologically trivial. The spectral structure in the background of the Fubini-Lipatov instanton can be continuously deformed into that in the flat background, establishing a one-to-one correspondence between the two spectra. However, when considering topologically nontrivial backgrounds for Yang-Mills and $\mathbb{CP}(N\!-\!1)$ theories, the spectrum undergoes restructuring. In these cases, a mismatch between the spectra around the instanton and the trivial vacuum occurs.

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arXiv2023

Consistency of $χ$SB in chiral Yang-Mills theory with adiabatic continuity

We study the pattern of chiral symmetry breaking ($χ$SB) in the $ψχη$ model (with the chiral fermion sector containing $ ψ^{\{ij\}}$, $χ_{[ij]}$, and $η_{i}^{A}$, see [1]) on $\mathbb{R}^3 \times S^{1}_{L}$ and derive implications to $\mathbb{R}^4$ physics. Center-symmetric vacua are stabilized by a double-trace deformation. With the center symmetry maintained at small $L(S^1)\ll Λ^{-1}$, i.e. at weak coupling, no phase transitions are expected in passing to large $L(S^1)\gg Λ^{-1}$ (here $Λ$ is the dynamical Yang-Mills scale). Starting with the small $L$-limit, we find the leading-order nonperturbative corrections in the given theory. The instanton-monopole operators induce the adjoint chiral condensate $\langle ψ^{\{ij\}}χ_{[jk]}\rangle \neq 0$ at weak coupling i.e. at $L(S^1)\ll Λ^{-1}$. Then adiabatic continuity tells us that $\langle ψ^{\{ij\}}χ_{[jk]}\rangle \neq 0$ exists on $\mathbb{R}^4$, in full accord with the prediction [2]. Simultaneously with $\langle ψ^{\{ij\}}χ_{[jk]}\rangle \sim Λ^3δ^i_k$ the SU($N_c$) gauge symmetry is spontaneously broken at strong coupling down to its maximal Abelian subgroup.

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arXiv2022

50 Years of Quantum Chromodynamics

This paper presents a comprehensive review of both the theory and experimental successes of Quantum Chromodynamics, starting with its emergence as a well defined theory in 1972-73 and following developments and results up to the present day. Topics include a review of the earliest theoretical and experimental foundations; the fundamental constants of QCD; an introductory discussion of lattice QCD, the only known method for obtaining exact predictions from QCD; methods for approximating QCD, with special focus on effective field theories; QCD under extreme conditions; measurements and predictions of meson and baryon states; a special discussion of the structure of the nucleon; techniques for study of QCD at high energy, including treatment of jets and showers; measurements at colliders; weak decays and quark mixing; and a section on the future, which discusses new experimental facilities or upgrades currently funded. The paper is intended to provide a broad background for Ph.D. students and postdocs starting their career. Some contributions include personal accounts of how the ideas or experiments were developed.

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arXiv2022

Infrared Renormalons in Supersymmetric Theories

I argue that in large-$N$ supersymmetric QCD infrared renormalons are absent in the conformal window, there is no need in conspiracy, and vacuum expectation values of at least some gluon operators vanish. Basing on this conclusion I conjecture that in supersymmetric gluodynamics (supersymmetric Yang-Mills theory without matter) at least the leading renormalon ambiguity disappears which would be consistent with the fact that the gluon condensate vanishes in this theory, $\langle G_{μν}^a G^{μν\,a}\rangle =0$.

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arXiv2022

Four-fermion deformations of the massless Schwinger model and confinement

We consider the massless charge-$N$ Schwinger model and its deformation with two four-fermion operators. Without the deformations, this model exhibits chiral symmetry breaking without confinement. It is usually asserted that the massless Schwinger model is always deconfined and a string tension emerges only when a mass for the fermion field is turned on. We show that in the presence of these four-fermion operators, the massless theory can in fact confine. One of the four-fermion deformations is chirally neutral, and is a marginal deformation. The other operator can be relevant or irrelevant, and respects a $\mathbb{Z}_2$ subgroup of chiral symmetry for even $N$, hence forbidding a mass term. When it is relevant, even the exactly massless theory exhibits both confinement and spontaneous chiral symmetry breaking. The construction is analogous to QCD(adj) in 2d. While the theory without four-fermion deformations is deconfined, the theory with these deformations is generically in a confining phase. We study the model on $\mathbb{R}^2$ using bosonization, and also analyze the mechanism of confinement on $\mathbb{R}\times S^1$, where we find that confinement is driven by fractional instantons.

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arXiv2022

The Inception, the Concept and the Second Life of Supersymmetry

I give a general non-technical review of supersymmetry and its modern applications in phenomenology and quantum field theories at strong coupling. Invited Talk at the conference {\sl Frontiers of Fundamental Physics, FFP16, May 23-26, 2022, Istanbul, Türkiye}

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