Large Limit Research Articles

A precision relation between Γ(K → μ+μ−)(t) and mathcal{B}left({K}_Lto {mu}^{+}{mu}^{-}right)/mathcal{B}left({K}_Lto gamma gamma right)

We find that the phase appearing in the unitarity relation between mathcal{B}left({K}_Lto {mu}^{+}{mu}^{-}right) and mathcal{B}left({K}_Lto gamma gamma right) is equal to the phase shift in the interference term of the time- dependent K → μ+μ− decay. A probe of this relation at future kaon facilities constitutes a Standard Model test with a theory precision of about 2%. The phase has further importance for sensitivity studies regarding the measurement of the time-dependent K → μ+μ− decay rate to extract the CKM matrix element combination mid {V}_{ts}{V}_{td}sin left(beta +{beta}_sright)mid approx {A}^2{lambda}^5overline{eta} . We find a model-independent theoretically clean prediction, cos2φ0 = 0.96 ± 0.03. The quoted error is a combination of the theoretical and experimental errors, and both of them are expected to shrink in the future. Using input from the large-NC limit within chiral perturbation theory, we find a theory preference towards solutions with negative cos φ0, reducing a four-fold ambiguity in the angle φ0 to a two-fold one.

Fusion hierarchies, T-systems and Y-systems for the dilute loop models on a strip

We study the dilute loop models on the geometry of a strip of width N. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its T- and Y-systems, for these four boundary conditions and with a generic crossing parameter λ. For rational and thus a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-N expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small N. The present work complements a previous study (Morin-Duchesne and Pearce 2019 J. Stat. Mech. 1909) that investigated the dilute loop models with periodic boundary conditions.

Isovector helicity quark quasi-distributions inside a large-Nc nucleon

In this letter, we report the results for the isovector polarized quark quasi-distribution functions Δu(x,v)−Δd(x,v) in the large-Nc limit, calculated within the chiral quark-soliton model. It is shown that the polarized quark quasi-distributions present good convergence to the light-cone PDFs in the limit of the nucleon boost momentum PN→∞ (or the velocity v→1), compared to the case of the unpolarized isosinglet distributions.

Random Euclidean coverage from within

Let X_1,X_2, ldots be independent random uniform points in a bounded domain A subset mathbb {R}^d with smooth boundary. Define the coverage thresholdR_n to be the smallest r such that A is covered by the balls of radius r centred on X_1,ldots ,X_n. We obtain the limiting distribution of R_n and also a strong law of large numbers for R_n in the large-n limit. For example, if A has volume 1 and perimeter |partial A|, if d=3 then mathbb {P}[npi R_n^3 - log n - 2 log (log n) le x] converges to exp (-2^{-4}pi ^{5/3} |partial A| e^{-2 x/3}) and (n pi R_n^3)/(log n) rightarrow 1 almost surely, and if d=2 then mathbb {P}[n pi R_n^2 - log n - log (log n) le x] converges to exp (- e^{-x}- |partial A|pi ^{-1/2} e^{-x/2}). We give similar results for general d, and also for the case where A is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform.

Cosmic F- and D-strings from pure Yang–Mills theory

We discuss the formation of cosmic strings or macroscopic color flux tubes after the deconfinement/confinement phase transition in the pure Yang–Mills theory. Based on holographic dual descriptions, these cosmic strings can be interpreted as fundamental (F-) strings or wrapped D-branes (which we call as D-strings) in the gravity side, depending on the structure of the gauge group. In fact, the reconnection probabilities of the F- and D-strings are suppressed by factors of 1/N2 and e−cN, where c=O(1), in a large-N limit, respectively. Supported by the picture of electric–magnetic duality, we discuss that color flux tubes form after the deconfinement/confinement phase transition, just like the formation of local cosmic strings after spontaneous symmetry breaking in the weak-U(1) gauge theory. We use an extended velocity-dependent one-scale model to describe the dynamics of the string network and calculate the gravitational wave signals from string loops. We also discuss the dependence on the size of produced string loops.

Vector bundles on fuzzy Kähler manifolds

Abstract We propose a matrix regularization of vector bundles over a general closed Kähler manifold. This matrix regularization is given as a natural generalization of the Berezin–Toeplitz quantization and gives a map from sections of a vector bundle to matrices. We examine the asymptotic behaviors of the map in the large-N limit. For vector bundles with algebraic structure, we derive a beautiful correspondence of the algebra of sections and the algebra of corresponding matrices in the large-N limit. We give two explicit examples for monopole bundles over a complex projective space CPn and a torus T2n.

Cardy expansion of 3d superconformal indices and corrections to the dual black hole entropy

We consider the superconformal index of three-dimensional mathcal{N} = 2 supersymmetric field theories computed via localization on S1 × S2. We systematically develop an expansion where the ratio of the radius of S1 to the radius of S2 is taken very small — a Cardy-like expansion. We emphasize the sub-leading structures in this Cardy-like expansion as well as their interplay with the large-N limit for theories with gauge group of the form product of U(N) factors. We demonstrate that taking the large-N limit first leads to an expression for the effective action yielding the index that only includes terms proportional to 1/β and powers of βi=0,1,2 where β is the ratio of radii. As we depart from the β → 0 limit for finite N, we find indications of non-perturbative contributions of the form e−1/β. We present an explicit evaluation of the superconformal index for the ABJM theory in the large-N limit that is exact in β up to non-perturbative corrections. For the ABJM theory we explore the implications of the Cardy-like expansion for corrections to the entropy of the rotating, electricallly charged, asymptotically AdS4 dual black hole. Interestingly, we find a closed form for the entropy that includes all perturbative corrections in β and takes the form of the leading order expression but with shifted charges.

Nematic phases and elastoresistivity from a multiorbital non-Fermi liquid

We propose and study a two-orbital lattice extension of the Sachdev-Ye-Kitaev model in the large-N limit. The phase diagram of this model features a high-temperature isotropic non-Fermi liquid which undergoes first-order thermal transition into a nematic insulator or continuous thermal transition into a nematic metal phase, separated by a tunable tricritical point. These phases arise from spontaneous partial orbital polarization of the multiorbital non-Fermi liquid. We explore the spectral and transport properties of this model, including d.c. elastoresistivity, which exhibits a peak near nematic transition, as well as nonzero frequency elastoconductivity. Our work offers a useful perspective on nematic phases and transport in correlated multiorbital systems.

Size-dependent diffusion of supported metal nanoclusters: mean-field-type treatments and beyond for faceted clusters.

Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, DN, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for DNversus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that DN ∼ hN-4/3neq. Here, h = ν exp[-Ed/(kBT)] denotes the hop rate, and neq = exp[-Eform/(kBT)] the density of those adatoms. The adatom formation energy, Eform, approaches a finite large-N limit, as does the effective barrier, Eeff = Ed + Eform, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of Eform and neqversus N for our model. MF theory then captures the occurrence of local minima in DNversus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an Eeff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment.

Pion spectroscopy and dynamics using the holographic light-front Schrödinger equation and the 't Hooft equation

We show that the holographic Schrödinger equation of light-front chiral QCD, together with the 't Hooft equation of (1+1)-dimensional QCD in the large Nc limit, can simultaneously describe pion spectroscopy as well as the pion decay constant, charge radius, electromagnetic form factor, photon-to-pion transition form factor, Parton Distribution Function (PDF) and Distribution Amplitude (DA). Furthermore, the chiral-limit constraints, as encoded in the Gell-Mann-Oakes-Renner (GMOR) relation, are satisfied.

Matrix regularization for tensor fields

Abstract We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices, and area-preserving diffeomorphisms and local rotations of the orthonormal frame are both realized as unitary similarity transformations of matrices in a unified way. We also show that the matrix commutator corresponds to the covariantized Poisson bracket for tensor fields in the large-N limit.

Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory

Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them in tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian distributions as the simplest case, and derive an explicit formula for signed distributions of real tensor eigenvectors: Each real tensor eigenvector contributes to the distribution by ±1, depending on the sign of the determinant of an associated Hessian matrix. The formula is expressed by the confluent hypergeometric function of the second kind, which is obtained by computing a partition function of a four-fermi theory. The formula can also serve as lower bounds of real eigenvector distributions (with no signs), and their tightness/looseness are discussed by comparing with Monte Carlo simulations. Large-N limits are taken with the characteristic oscillatory behavior of the formula being preserved.

The large-N limit of 4d superconformal indices for general BPS charges

We study the superconformal index of mathcal{N} = 1 quiver theories at large-N for general values of electric charges and angular momenta, using both the Bethe Ansatz formulation and the more recent elliptic extension method. We are particularly interested in the case of unequal angular momenta, J1 ≠ J2, which has only been partially considered in the literature. We revisit the previous computation with the Bethe Ansatz formulation with generic angular momenta and extend it to encompass a large class of competing exponential terms. In the process, we also provide a simplified derivation of the original result. We consider the newly-developed elliptic extension method as well; we apply it to the J1 ≠ J2 case, finding a good match with the Bethe Ansatz results. We also investigate the relation between the two different approaches, finding in particular that for every saddle of the elliptic action there are corresponding terms in the Bethe Ansatz formula that match it at large-N.

The LPM effect in sequential bremsstrahlung: incorporation of “instantaneous” interactions for QCD

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We continue study of the case when the coherence lengths (formation lengths) of two consecutive splitting processes overlap, avoiding soft-emission approximations. Previous work made a “nearly-complete” calculation of the effect of overlapping formation times on gluonic splittings such as g → gg → ggg (with simplifying assumptions such as an infinite QCD medium and the large-Nc limit). In this paper, we extend those previous rate calculations from nearly-complete to complete by including processes involving the exchange of longitudinally-polarized gluons. In the context of Lightcone Pertubation Theory, used earlier for the “nearly-complete” calculation, such exchanges are instantaneous in lightcone time and have their own diagrammatic representation.

T-odd leading-twist quark TMDs at small x

We study the small-x asymptotics of the flavor non-singlet T-odd leading-twist quark transverse momentum dependent parton distributions (TMDs), the Sivers and Boer-Mulders functions. While the leading eikonal small-x asymptotics of the quark Sivers function is given by the spin-dependent odderon [1, 2], we are interested in revisiting the sub-eikonal correction considered by us earlier in [3]. We first simplify the expressions for both TMDs at small Bjorken x and then construct small-x evolution equations for the resulting operators in the large-Nc limit, with Nc the number of quark colors. For both TMDs, the evolution equations resum all powers of the double-logarithmic parameter αs ln2(1/x), where αs is the strong coupling constant, which is assumed to be small. Solving these evolution equations numerically (for the Sivers function) and analytically (for the Boer-Mulders function) we arrive at the following leading small-x asymptotics of these TMDs at large Nc:f1T⊥NSx≪1kT2=COxkT21x+C1xkT21x3.4αsNc4πh1⊥NSx≪1kT2=CxkT21x−1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\displaystyle \\begin{array}{l}{f}_{1T}^{\\perp NS}\\left(x\\ll 1,{k}_T^2\\right)={C}_O\\left(x,{k}_T^2\\right)\\frac{1}{x}+{C}_1\\left(x,{k}_T^2\\right){\\left(\\frac{1}{x}\\right)}^{3.4\\sqrt{\\frac{\\alpha_s{N}_c}{4\\pi }}}\\\\ {}{h}_1^{\\perp \ extrm{NS}}\\left(x\\ll 1,{k}_T^2\\right)=C\\left(x,{k}_T^2\\right){\\left(\\frac{1}{x}\\right)}^{-1}.\\end{array}} $$\\end{document}The functions CO(x, {k}_T^2 ), C1(x, {k}_T^2 ), and C(x, {k}_T^2 ) can be readily obtained in our formalism: they are mildly x-dependent and do not strongly affect the power-of-x asymptotics shown above. The function CO, along with the 1/x factor, arises from the odderon exchange. For the sub-eikonal contribution to the quark Sivers function (the term with C1), our result shown above supersedes the one obtained in [3] due to the new contributions identified recently in [4].

From Skyrmions to One Flavored Baryons and Beyond

While the identification of skyrmions as the low energy description of baryons in Nf≥2 QCD is known for decades, a parallel construction for the case of Nf=1 is more mysterious. In the case of one fermionic flavor, there is no chiral symmetry breaking, no non-linear sigma model, and the conventional construction of skyrmions fails to work. In this article, I will review developments from the last couple of years trying to identify baryons as certain singular configurations in the large Nc limit of Nf=1 QCD. We will give various arguments supporting this identification, and discuss some of its applications. Unlike skyrmions, the new baryons are not contained completely inside the low energy effective theory. They give rise to a singular ring on which the chiral condensate must vanish, with new degrees of freedom living on this ring. These configurations may serve as a bridge between the UV and the IR, and hopefully shed some light on the connection between different phases of QCD.

Incompleteness of the large-N analysis of the O(N) models: Nonperturbative cuspy fixed points and their nontrivial homotopy at finite N.

We summarize the usual implementations of the large-N limit of O(N) models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit N→∞. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as N increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of N. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all N>0 and show that they play an important role for the tricritical physics of O(N) models. Finally, we show that some of these fixed points are bivalued when they are considered as functions of d and N thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at N=∞ and d=3 is shown to play a crucial role for the internal consistency of all our results.

Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

An exact map was established by Lacroix-A-Chez-Toine et al. in (Phys Rev A 99(2):021602, 2019) between the N complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions of N non-interacting Fermions in a rotating trap in the ground state. An important quantity is the statistics of the number of Fermions mathcal {N}_a in a disc of radius a. Extending the work (Lacroix-A-Chez-Toine et al., in Phys Rev A 99(2):021602, 2019) covering Gaussian and rotationally invariant potentials Q, we present a rigorous analysis in planar complex and symplectic ensembles, which both represent 2D Coulomb gases. We show that the variance of mathcal {N}_a is universal in the large-N limit, when measured in units of the mean density proportional to Delta Q, which itself is non-universal. This holds in the large-N limit in the bulk and at the edge, when a finite fraction or almost all Fermions are inside the disc. In contrast, at the origin, when few eigenvalues are contained, it is the singularity of the potential that determines the universality class. We present three explicit examples from the Mittag-Leffler ensemble, products of Ginibre matrices, and truncated unitary random matrices. Our proofs exploit the integrable structure of the underlying determinantal respectively Pfaffian point processes and a simple representation of the variance in terms of truncated moments at finite-N.

Semiflexible polymer solutions. II. Fluctuations and Frank elastic constants.

We study the collective elastic behavior of semiflexible polymer solutions in a nematic liquid-crystalline state using polymer field theory. Our polymer field-theoretic model of semiflexible polymer solutions is extended to include second-order fluctuation corrections to the free energy, permitting the evaluation of the Frank elastic constants based on orientational order fluctuations in the nematic state. Our exact treatment of wormlike chain statistics permits the evaluation of behavior from the nematic state, thus accurately capturing the impact of single-chain behavior on collective elastic response. Results for the Frank elastic constants are presented as a function of aligning field strength and chain length, and we explore the impact of conformation fluctuations and hairpin defects on the twist, splay, and bend moduli. Our results indicate that the twist elastic constant Ktwist is smaller than both bend and splay constants (Kbend and Ksplay, respectively) for the entire range of polymer rigidity. Splay and bend elastic constants exhibit regimes of dominance over the range of chain stiffness, where Ksplay > Kbend for flexible polymers (large-N limit) while the opposite is true for rigid polymers. Theoretical analysis also suggests the splay modulus tracks exactly to that of the end-to-end distance in the transverse direction for semiflexible polymers at intermediate to large-N. These results provide insight into the role of conformation fluctuations and hairpin defects on the collective response of polymer solutions.

Referral-marketing practices of travel agencies

Organisations can make use of high levels of customer-service delivery to stimulate positiveword-of-mouth referrals. Referral marketing is primarily applied by organisations when the budgetis the marketers’ largest limitation. The purpose of the article is to indicate the current status of therelationships that travel agencies in the Western Cape Province have with their referral market,and to provide recommendations to the management of travel agencies in said province regardingthe improved application of the principles of relationship marketing to this market. The targetpopulation for this study comprised 118 travel agencies of which 61 managers and/or ownersparticipated through personal interviews in the completion of questionnaires. Data analysis wasdone by calculating averages and standard deviations, explorative factor analysis, CronbachAlpha-values and practical significance by means of effect sizes. The findings stipulate that theowners and managers of travel agencies must create and establish more closely integratedrelationships with strategic suppliers to increase the value offering of their products and servicesto customers, whereby the travel agencies can ensure that the needs and wants of customerscan be satisfied more successfully, which, in the long run, could lead to positive word-of-mouthreferrals.