Sigma Model Research Articles

Entanglement and Rényi entropies of (1+1)-dimensional O(3) nonlinear sigma model with tensor renormalization group

We investigate the entanglement and Rényi entropies for the (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method. The central charge is determined from the asymptotic scaling properties of both entropies. We also examine the consistency between the entanglement entropy and the nth-order Rényi entropy with n → 1.

Axio-Chameleons: a novel string-friendly multi-field screening mechanism

Scalar-tensor theories with the shift symmetries required by light scalars are well-explored modifications to GR. For these, two-derivative scalar self-interactions usually dominate at low energies and interestingly compete with the two-derivative metric interactions of GR itself. Although much effort has been invested in single scalars (on grounds of simplicity) these happen to have no two-derivative interactions, requiring such models to explore higher-derivative interactions (that usually would be less important at low-energies). This suggests multiple-scalar sigma models as well-motivated candidates for finding new phenomena in tests of gravity. We identify a new multi-field screening mechanism appropriate for two light scalar fields (an axion and a Brans-Dicke style dilaton) that relies on their mutual two-derivative interactions. We show how very weak axion-matter couplings can introduce axion gradients that can reduce the apparent coupling of the Brans-Dicke scalar to macroscopic matter sources. We further identify a relaxation mechanism that allows this reduction to be amplified to a suppression by the ratio of the axion gradient's length scale to the source's radius (similar in size to the suppression found in Chameleon models). Unlike some screening mechanisms our proposal is technically natural and works deep within the regime of control of the low-energy EFT. It uses only ingredients that commonly appear in the low-energy limit of string vacua and so is likely to have wider applications to models that admit UV completions. We briefly discuss phenomenological implications and challenges for this scenario, which suggests re-examination of decay loss bounds and the value of equivalence-principle tests for different-sized objects.

Moduli spaces of instantons in flag manifold sigma models. Vortices in quiver gauge theories

In this paper, we discuss lumps (sigma model instantons) in flag manifold sigma models. In particular, we focus on the moduli space of BPS lumps in general Kähler flag manifold sigma models. Such a Kähler flag manifold, which takes the form Un1+⋯+nL+1Un1×⋯×UnL+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\ extrm{U}\\left({n}_1+\\cdots +{n}_{L+1}\\right)}{\ extrm{U}\\left({n}_1\\right)\ imes \\cdots \ imes \ extrm{U}\\left({n}_{L+1}\\right)} $$\\end{document}, can be realized as a vacuum moduli space of a U(N1) × ··· × U(NL) quiver gauged linear sigma model. When the gauge coupling constants are finite, the gauged linear sigma model admits BPS vortex configurations, which reduce to BPS lumps in the low energy effective sigma model in the large gauge coupling limit. We derive an ADHM-like quotient construction of the moduli space of BPS vortices and lumps by generalizing the quotient construction in U(N) gauge theories by Hanany and Tong. As an application, we check the dualities of the 2d models by computing the vortex partition functions using the quotient construction.

Gravitational waves from composite dark sectors

We study under which conditions a first-order phase transition in a composite dark sector can yield an observable stochastic gravitational-wave signal. To this end, we employ the Linear-Sigma model featuring Nf = 3, 4, 5 flavours and perform a Cornwall-Jackiw-Tomboulis computation also accounting for the effects of the Polyakov loop. The model allows us to investigate the chiral phase transition in regimes that can mimic QCD-like theories incorporating in addition composite dynamics associated with the effects of confinement-deconfinement phase transition. A further benefit of this approach is that it allows to study the limit in which the effective interactions are weak. We show that strong first-order phase transitions occur for weak effective couplings of the composite sector leading to gravitational-wave signals potentially detectable at future experimental facilities.

Gravitational wave footprints from Higgs-portal scalegenesis with multiple dark chiral scalars* *Supported in part by the National Science Foundation of China (NSFC) (11747308, 11975108, 12047569), the Seeds Funding of Jilin University (S.M.), and Toyama First Bank, Ltd (H.I.)

We discuss the gravitational wave (GW) spectra predicted from the electroweak scalegenesis of the Higgs portal type with a large number of dark chiral flavors, which many flavor QCD would underlie and give the dynamical explanation of the negative Higgs portal coupling required to trigger the electroweak symmetry breaking. We employ the linear-sigma model as the low-energy description of dark many flavor QCD and show that the model undergoes ultra-supercooling due to the produced strong first-order thermal phase transition along the (approximately realized) flat direction based on the Gildener-Weinberg mechanism. Passing through evaluation of the bubble nucleation/percolation, we address the reheating and relaxation processes, which are generically non-thermal and nonadiabatic. Parametrizing the reheating epoch in terms of the e-folding number, we propose proper formulae for the redshift effects on the GW frequencies and signal spectra. It then turns out that the ultra-supercooling predicted from the Higgs-portal scalegenesis generically yields none of GW signals with the frequencies as low as nano Hz, unless the released latent heat is transported into another sector other than reheating the universe. Instead, models of this class prefer to give the higher frequency signals and still keeps the future prospected detection sensitivity, like at LISA, BBO, and DECIGO, etc. We also find that with large flavors in the dark sector, the GW signals are made further smaller and the peak frequencies higher. Characteristic phenomenological consequences related to the multiple chiral scalars include the prediction of dark pions with the mass much less than TeV scale, which is also briefly addressed.

Lifting Klein-Gordon/Einstein solutions to general nonlinear sigma-models: the wormhole example

We describe a simple technique for generating solutions to the classical field equations for an arbitrary nonlinear sigma-model minimally coupled to gravity. The technique promotes an arbitrary solution to the coupled Einstein/Klein-Gordon field equations for a single scalar field σ to a solution of the nonlinear sigma-model for N scalar fields minimally coupled to gravity. This mapping between solutions does not require there to be any target-space isometries and exists for every choice of geodesic computed using the target-space metric. In some special situations — such as when the solution depends only on a single coordinate (e.g. for homogeneous time-dependent or static spherically symmetric configurations) — the general solution to the sigma-model equations can be obtained in this way. We illustrate the technique by applying it to generate Euclidean wormhole solutions for multi-field sigma models coupled to gravity starting from the simplest Giddings-Strominger wormhole, clarifying why in the wormhole case Minkowski-signature target-space geometries can arise. We reproduce in this way the well-known axio-dilaton string wormhole and we illustrate the power of the technique by generating simple perturbations to it, like those due to string or α′ corrections.

Sigma model instantons and singular tau function

The generating series for the instanton contribution to Green functions of the $2D$ sigma model was found in the works of Schwarz, Fateev and Frolov. We show that this series can be written as a formal tau function of the two-sided two-component KP hierarchy. We call it formal singular tau function because this tau function is a sum where each term is the infrared and ultraviolet divergent one exactly as the series found by the mentioned authors. However one can regularize this singluar tau function and to obtain regular observables. This is because observables contains ratious of mentioned divergent expressions. Thus, we enladge the families of tau functions to work with.

Quantum cohomology from mixed Higgs-Coulomb phases

We generalize Coulomb-branch-based gauged linear sigma model (GLSM)–computations of quantum cohomology rings of Fano spaces. Typically such computations have focused on GLSMs without superpotential, for which the low energy limit of the GLSM is a pure Coulomb branch, and quantum cohomology is determined by the critical locus of a twisted one-loop effective superpotential. We extend these results to cases for which the low energy limit of the GLSM includes both Coulomb and Higgs branches, where the latter is a Landau-Ginzburg orbifold. We describe the state spaces and products of corresponding operators in detail, comparing a geometric phase description, where the operator product ring is quantum cohomology, to the description in terms of Coulomb and Higgs branch states. As a concrete test of our methods, we compare to existing mathematics results for quantum cohomology rings of hypersurfaces in projective spaces.

Conformal perturbation theory on K3: the quartic Gepner point

The Gepner model (2)4 describes the sigma model of the Fermat quartic K3 surface. Moving through the nearby moduli space using conformal perturbation theory, we investigate how the conformal weights of its fields change at first and second order and find approximate minima. This serves as a toy model for a mechanism that could produce new chiral fields and possibly new nearby rational CFTs.

Quantum Energy Inequalities in Integrable Models with Several Particle Species and Bound States

We investigate lower bounds to the time-smeared energy density, so-called quantum energy inequalities (QEI), in the class of integrable models of quantum field theory. Our main results are a state-independent QEI for models with constant scattering function and a QEI at one-particle level for generic models. In the latter case, we classify the possible form of the stress-energy tensor from first principles and establish a link between the existence of QEIs and the large-rapidity asymptotics of the two-particle form factor of the energy density. Concrete examples include the Bullough–Dodd, the Federbush, and the O(n)-nonlinear sigma models.

Note on NLSM tree amplitudes and soft theorems

AbstractThis note provides a new point of view for bootstrapping the tree amplitudes of the nonlinear sigma model (NLSM). We use the universality of single soft behavior, together with the double copy structure, to completely determine the tree amplitudes of the NLSM. We first observe Adler’s zero for four-point NLSM amplitudes, by considering kinematics. Then we assume the universality of Adler’s zero and use this requirement to construct general tree amplitudes of the NLSM in the expanded formula, i.e., the formula of expanding NLSM amplitudes to bi-adjoint scalar amplitudes, which allows us to give explicit expressions of amplitudes with arbitrary numbers of external legs. The construction does not require the assumption of quartic diagrams. We also derive double soft factors for NLSM tree amplitudes based on the resulting expanded formula, and the results are consistent with those in the literature.

Modified supersymmetric indices in AdS3/CFT2

We consider the \U0001d4a9 = (2, 2) AdS3/CFT2 dualities proposed by Eberhardt, where the bulk geometry is AdS3 × (S3 × T4)/ℤk, and the CFT is a deformation of the symmetric orbifold of the supersymmetric sigma model T4/ℤk (with k = 2, 3, 4, 6). The elliptic genera of the two sides vanish due to fermionic zero modes, so for microstate counting applications one must consider modified supersymmetric indices. In an analysis similar to that of Maldacena, Moore, and Strominger for the standard \U0001d4a9 = (4, 4) case of T4, we study the appropriate helicity-trace index of the boundary CFTs. We encounter a strange phenomenon where a saddle-point analysis of our indices reproduces only a fraction (respectively 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}, 23\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{2}{3} $$\\end{document}, 34\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{3}{4} $$\\end{document}, 56\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{5}{6} $$\\end{document}) of the Bekenstein-Hawking entropy of the associated macroscopic black branes.

The O(N) monolith reloaded: sum rules and Form Factor Bootstrap

We revisit the space of gapped quantum field theories with a global O(N) symmetry in two spacetime dimensions. Previous works using S-matrix bootstrap revealed a rich space in which integrable theories such as the non-linear sigma model appear at special points on the boundary, along with an abundance of unknown models hinting at a non conventional UV behaviour. We extend the S-matrix set-up by including into the bootstrap form factors and spectral functions for the stress-energy tensor and conserved O(N) currents. Sum rules allow us to put bounds on the central charges of the conformal field theory (CFT) in the UV. We find that a big portion of the boundary can only flow from CFTs with infinite central charges. We track this result down to a particular behaviour of the amplitudes in physical kinematics and discuss its physical implications.

Even-point multi-loop unitarity and its applications: exponentiation, anomalies and evanescence

We identify novel structure in newly computed multi-loop amplitudes and quantum actions for even-point effective field theories, including both the nonlinear sigma model (NLSM) and double-copy gauge theories such as Born-Infeld and its supersymmetric generalizations. We exploit special properties of all even-point theories towards establishing an efficient unitarity based amplitude construction. In doing so, we find evidence that the leading IR divergence of NLSM amplitudes exponentiates when the target space is CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{CP} $$\\end{document}1 ≅ SU(2)/U(1). We then systematically compute the two-loop anomalous behavior of Born-Infeld, and find that the counterterms needed to restore U(1) invariant behavior at loop-level can be constructed via a symmetric-structure double-copy. We also demonstrate that the divergent part of the one-minus (−+++) two-loop anomaly vanishes upon introducing an evanescent operator. In addition to these purely photonic counterterms, we verify through explicit calculation that the anomalous matrix elements that violate U(1) duality invariance can be alternatively cancelled by summing over internal N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 DBIVA superfields. Finally we find that N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Dirac-Born-Infeld-Volkov-Akulov (DBIVA) amplitudes admit double-copy construction through two-loop order by reproducing our unitarity based result with a double copy between color-dual N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills and our two-loop NLSM amplitudes. This result supports the possibility of color-dual representations for NLSM beyond one-loop. We conclude with an overview of how D-dimensional four-photon counterterms can be constructed in generality with the symmetric-structure double-copy, and outline a convenient way of counting evanescent operators using Hilbert series as generating functions.

Development of a Lean Six Sigma Model and Resilient Supply Chain Using Modern Fuzzy Inference Approaches and Design of Experiments (Case Study: Asia Factory)

Objective: The present research aims to develop a Lean Six Sigma model and a resilient supply chain. Methodology: To develop the Lean Six Sigma model and resilient supply chain using a grounded theory method, interviews with experts were conducted to examine the factors influencing the model's development. Then, the Delphi process and surveys from relevant experts (faculty members, managers of food companies, and supply chain experts) were used to finalize the list of dimensions and components via the grounded theory method and to determine the importance coefficient of each. Finally, the identified input variables were converted into explicit or numerical values using fuzzy inference algorithms by a defuzzification unit. Using the design of experiments technique, the 3^(k-p) method was applied to express the impact of each influencing factor on the output characteristics as a model. Considering the qualitative nature of the research method, the data obtained from the study were analyzed using NVIVO version 11 software in the first step. In the next step, MATLAB software was used for the fuzzy inference section, and finally, MINITAB software was used in the design of experiments section. Findings: The results showed that all input variables affecting flexibility, capacity, and adaptability were significant at a 95% confidence level. Additionally, Total Quality Management (TQM) had the most significant impact on the variables of flexibility and capacity, while Quality Function Deployment (QFD) had the most significant impact on adaptability. Conclusion: The proposed model, aiming for integrated information across the food chain, spans from primary agricultural production to the consumer, replacing human mechanisms and potential errors.

Utilizing Lean Six Sigma and Waste Assessment Model to Reduce Waste in the Hot Rolled Coil Production

The global steel industry is pivotal, driven by substantial demand. In Indonesia, a leading steel manufacturing company, boasting significant production capacities for Hot Rolled Coil (HRC) and Cold Rolled Coil (CRC) products, with capacities of 2,400,000 and 850,000 tons per year. However, current production lags at 1,500,000 and 500,000 tons annually for HRC and CRC. Notably, the production process recorded defect ratios of 15.85%, 15.65%, and 15.56% from 2020 to 2022, with waste waiting accounting for a breakdown time and operating time ratio of ±40%. The study employs Lean Six Sigma to analyze and mitigate waste during HRC production, aiming to augment product value, production efficiency, and cost reduction. Through Value Stream Mapping and Process Activity Mapping, 44.93% of Necessary Not Value- Added Activities (NNVA). In the next stage, critical wastes including waste defects, overproduction, and waiting are selected using the Waste Assessment Model. The Analysis stage employs fishbone diagrams and Failure Mode and Effect Analysis to determine root causes and assess risk. Proposed improvements include toolbox meetings, maintenance scheduling, check sheets, a work roll replacement algorithm, and sensor adjustments, projected to raise sigma values significantly for waste defects, overproduction, and waiting.

Grassmannian Sigma Models

Grassmannian Sigma Models

Approximation of BPS Skyrme model using modified Lagrangian Skyrmion

One of the nuclear atomic models represented by Skyrmion was the Skyrme model. This model was a modified nonlinear sigma model with a Skyrme field where the classical solution use generalized sixth order terms and potential terms. The binding energy that will be studied in the Skyrme SU(2) model is to generalize the second order nonlinear sigma model terms with sixth order derivative terms. The Lagrangian will be obtained for these two terms to find the BPS (Bogomolny Prasad Sommerfield) solution for the profile function numerically. The result of numerical calculation will be used to calculate static energy and rotational energy, where the characteristic of the nucleus can be observed from these two energies. Furthermore, the value of the coupling constant in the Lagrangian Skyrmion will be calculated from the static energy and rotational energy obtained previously. These values are expected to help in the application of Skyrme model for many research physics field.

QCD-collapsed domain walls: QCD phase transition and gravitational wave spectroscopy

For a discrete symmetry that is anomalous under QCD, the domain walls produced in the early universe from its spontaneous breaking can naturally annihilate due to QCD instanton effects. The gravitational waves generated from wall annihilation have their amplitude and frequency determined by both the discrete symmetry breaking scale and the QCD scale. The evidence of stochastic gravitational waves at nanohertz observed by pulsar timing array experiments suggests that the discrete-symmetry-breaking scale is around 100 TeV, assuming the domain-wall explanation. The annihilation temperature is about 100 MeV, which could naturally be below the QCD phase transition temperature. We point out that the QCD phase transition within some domains with an effective large QCD θ angle could be a first-order one. To derive the phase diagram in θ and temperature, we adopt a phenomenological linear sigma model with three quark flavors. The domain-wall explanation for the NANOGrav, EPTA, PPTA and CPTA results hints at a first-order QCD phase transition, which predicts additional gravitational waves at higher frequencies. If the initial formation of domain walls is also a first-order process, this class of domain-wall models predicts an interesting gravitational wave spectroscopy with frequencies spanning more than ten orders of magnitude, from nanohertz to 100 Hz.

Longitudinal Galilean and Carrollian limits of non-relativistic strings

It is well known that one can take an infinite speed of light limit that gives rise to non-relativistic strings with a relativistic worldsheet sigma model but with a non-relativistic target space geometry. In this work we systematically explore two further limits in which the worldsheet becomes non-Lorentzian. The first gives rise to a Galilean string with a Galilean structure on the worldsheet, extending previous work on Spin Matrix-related string theory limits. The second is a completely novel limit leading to a worldsheet theory with a Carrollian structure. We find the Nambu-Goto and Polyakov formulations of both limits and explore gauge fixing choices. Furthermore, we study in detail the case of the Galilean string for a class of target space geometries that are related to Spin Matrix target space geometries, for which the Nambu-Goto action (in static gauge) is quadratic in the fields.