Large Limit Research Articles

The Brown measure of the free multiplicative Brownian motion

The free multiplicative Brownian motion b_{t} is the large-N limit of the Brownian motion on mathsf {GL}(N;mathbb {C}), in the sense of *-distributions. The natural candidate for the large-N limit of the empirical distribution of eigenvalues is thus the Brown measure of b_{t}. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region Sigma _{t} that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density W_{t} on overline{Sigma }_{t}, which is strictly positive and real analytic on Sigma _{t}. This density has a simple form in polar coordinates: Wt(r,θ)=1r2wt(θ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} W_{t}(r,\ heta )=\\frac{1}{r^{2}}w_{t}(\ heta ), \\end{aligned}$$\\end{document}where w_{t} is an analytic function determined by the geometry of the region Sigma _{t}. We show also that the spectral measure of free unitary Brownian motion u_{t} is a “shadow” of the Brown measure of b_{t}, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

A theoretical interpretation of the anomalous reduced E2 transition probabilities along the yrast line of neutron-deficient nuclei

The anomalous low-energy E2-related behavior of a triaxially deformed nuclei is examined within the framework of the interacting boson model. The results show striking features that include a B(E2;41+→21+)/B(E2;21+→01+)<1.0 transition rate and a E(41+)/E(21+)>2.0 excitation energy ratio that can be tracked back to a finite-N effect, which in a large-N limit of the theory yields normal results for a stable γ-deformation. This description is shown to explain experimentally observed results in 172Pt and 168Os, and in doing so yields a deeper understanding of the physical features of a soft triaxially-deformed nucleus.

The LPM effect in sequential bremsstrahlung: 1/ {mathrm{N}}_{mathrm{c}}^2 corrections

An important question concerning in-medium high-energy parton showers in a quark-gluon plasma or other QCD medium is whether consecutive splittings of the partons in a given shower can be treated as quantum mechanically independent, or whether the formation times for two consecutive splittings instead have significant overlap. Various previous calculations of the effect of overlapping formation times have either (i) restricted attention to a soft bremsstrahlung limit, or else (ii) used the large-Nc limit (where Nc = 3 is the number of quark colors). In this paper, we make a first study of the accuracy of the large-Nc limit used by those calculations of overlap effects that avoid a soft bremsstrahlung approximation. Specifically, we calculate the 1/ {N}_{mathrm{c}}^2 correction to previous Nc = ∞ results for overlap g → gg → ggg of two consecutive gluon splittings g → gg. At order 1/ {N}_{mathrm{c}}^2 , there is interesting and non-trivial color dynamics that must be accounted for during the overlap of the formation times.

Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering

We present a comprehensive study of high-energy double logarithms in inclusive DIS. They appear parametrically as {alpha}_s^n ln2n−kx at the n-th order in perturbation theory in the splitting functions for the parton evolution and the coefficient functions for the hard scattering process, and represent the leading corrections at small x in the flavour non-singlet case. We perform their resummation, in terms of modified Bessel functions, to all orders in full QCD up to NNLL accuracy, and partly to N3LL and beyond in the large-nc limit, and provide fixed-order expansions up to five loops. In the flavour-singlet sector, where these double logarithms are sub-dominant at small x compared to single-logarithmic {alpha}_s^n x−1 lnn−kx BFKL contributions, we construct fixed-order expansions up to five loops at NNLL accuracy in full QCD. The results elucidate the analytic small-x structure underlying inclusive DIS results in fixed-order perturbation theory and provide important information for present and future numerical and analytic calculations of these quantities.

Definition and evolution of transverse momentum dependent distribution of twist-three

We present an in-depth analysis of transverse momentum dependent (TMD) distributions of twist-three. In particular, we focus on evolution equations, symmetry relations, parameterization, interpretation, small-b asymptotic behaviour and the structure of singularities. The starting point of discussion are the correlators with the definite TMD-twist. By considering suitable combinations of these correlators, we introduce physical TMD distribution of twist-three that can be used for practical applications. We also establish relations with generic TMD distribution of twist-three, and demonstrate that their evolution equations are autonomous in the large-Nc limit.

Large-time behaviour and the second eigenvalue problem for finite-state mean-field interacting particle systems

AbstractThis article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order $\exp\{N\Lambda\}$ for a suitable constant $\Lambda &gt; 0$ , the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{{-}N\Lambda\}$ . The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

Dark stress for improved lipid quantity and quality in bioprospected acid-tolerant green microalgae.

The cost of microalgae cultivation is one of the largest limitations to achieving sustainable, large-scale microalgae production of commercially desirable lipids. Utilizing CO2 as a 'free' carbon source from waste industrial flue gas emissions can offer wide-ranging cost savings. However, these gas streams typically create acidic environments, in which most microalgae cannot survive due to the concentration of CO2 and the presence of other acidic gasses such as NO2 and SO2. To address this situation, we investigated growth of a mixed acid-tolerant green microalgal culture (91% dominated by a single Coccomyxa sp. taxon) bioprospected at pH 2.8 from an acid mine drainage impacted water body. The culture was grown at pH 2.5 and fed with a simulated flue gas containing 6% CO2 and 94% N2. On reaching the end of the exponential growth phase, the culture was exposed to either continued light-dark cycle conditions or continual dark conditions. After three days in the dark, the biomass consisted of 28% of lipids, which was 42% higher than at the end of the exponential phase and 55% higher than the maximum lipid content achieved under light/dark conditions. The stress caused by being continually in the dark also favoured the production of omega-3 and omega-6 polyunsaturated fatty acids (PUFAs; 19.47% and 21.04%, respectively, after 7 days) compared to 7-days of light-dark treatment (1.94% and 9.53%, respectively) and showed an increase in nitrogen content (C:N ratio of 6.4) compared to light-dark treatment (C:N ratio of 11.9). The results of the research indicate that use of acid tolerant microalgae overcomes issues using flue gasses that will create an acidic environment and that applying dark stress is a low-cost stressor stimulates production of desirable dietary lipids.

Four-point functions with multi-cycle fields in symmetric orbifolds and the D1-D5 CFT

We study SN-invariant four-point functions with two generic multi-cycle fields and two twist-2 fields, at the free orbifold point of the D1-D5 CFT. We derive the explicit factorization of these functions following from the action of the symmetric group on the composite multi-cycle fields. Apart from non-trivial symmetry factors that we compute, the function with multi-cycle operators is reduced to a sum of connected correlators in which the composite fields have, at most, two cycles. The correlators with two double-cycle and two single-cycle fields give the leading order contribution in the large-N limit. We derive explicit formulas for these functions, encompassing a large class of choices for the single- and the double-cycle fields, including generic Ramond ground states, NS chiral fields and the marginal deformation operator. We are thus able to extract important dynamical information from the short-distance OPEs: conformal dimensions, R-charges and structure constants of families of BPS and non-BPS fields present in the corresponding light-light and heavy-light channels. We also discuss properties of generic multi-cycle Q-point functions in MN/SN orbifolds, using a technology due to Pakman, Rastelli and Razamat.

Interrelation of nonclassicality conditions through stabiliser group homomorphism

In this paper, we show that coherence witness for a single qubit itself yields conditions for nonlocality and entanglement inequalities for multiqubit systems. It also yields a condition for quantum discord in two-qubit systems. It is shown by employing homomorphism among the stabiliser group of a single qubit and those of multi-qubit states. Interestingly, globally commuting homomorphic images of single-qubit stabilisers do not allow for consistent assignments of outcomes of local observables. We employ these observables for the construction of conditions for nonlocality, entanglement, and quantum discord. As an application, we show that CHSH inequality can be straightforwardly generalised to nonlocality inequalities for multiqubit GHZ states. It also reconfirms the fact that quantumness prevails even in the large N-limit if coherence is sustained. The mapping provides a way to construct many nonlocality inequalities, given a seed inequality. This study gives us a motivation to gain better control over multiple degrees of freedom and multi-party systems. It is because, in multi-party systems, the same nonclassical feature, viz, coherence may appear in many avatars.

Functional renormalization group for multilinear disordered Langevin dynamics I Formalism and first numerical investigations at equilibrium

This paper aims at using the functional renormalization group formalism to study the equilibrium states of a stochastic process described by a quench–disordered multilinear Langevin equation. Such an equation characterizes the evolution of a time-dependent N-vector q(t) = {q 1(t), ⋯ q N (t)} and is traditionally encountered in the dynamical description of glassy systems at and out of equilibrium through the so-called Glauber model. From the connection between Langevin dynamics and quantum mechanics in imaginary time, we are able to coarse-grain the path integral of the problem in the Fourier modes, and to construct a renormalization group flow for effective Euclidean action. In the large N-limit we are able to solve the flow equations for both matrix and tensor disorder. The numerical solutions of the resulting exact flow equations are then investigated using standard local potential approximation, taking into account the quench disorder. In the case where the interaction is taken to be matricial, for finite N the flow equations are also solved. However, the case of finite N and taking into account the non-equilibrum process will be considered in a companion investigation.

Universality (beyond leading log) of soft radiative corrections to hat{q} in p⊥ broadening and energy loss

It has been known for many years that soft radiation can give potentially large double-logarithm corrections to p⊥ broadening of a high-energy particle traveling through QCD matter, but that this soft radiation correction can be absorbed into an effective value hat{q} eff for the medium p⊥-broadening parameter hat{q} . Here “soft” means high energy compared to medium scales but soft compared to the original high-energy particle traveling through the medium. A similar situation arises in the case of soft corrections to hard splitting of a high-energy particle, such as hard g→gg, where double logarithms can also be absorbed using the same effective hat{q} eff. In this paper, I study whether the same holds true for potentially large, subleading, single-logarthim corrections. The correspondence is more indirect for single logarithms, but I show (in the large-Nc limit) that single logarithms from soft radiation in the case of p⊥ broadening also determine the single logarithms from soft radiative corrections to hard g→gg splitting. Along the way, there is an interesting variation of the original BDMPS-Z calculation of splitting rates in the hat{q} approximation. I also discuss how, for soft-radiative corrections to hard splitting processes, there are two different types of “ hat{q} eff” that come into play, which differ by “iπ” terms that multiply single logarithms.

Higher-order non-global logarithms from jet calculus

Non-global QCD observables are characterised by a sensitivity to the full angular distribution of soft radiation emitted coherently in hard scattering processes. This complexity poses a challenge to their all-order resummation, that was formulated at the leading-logarithmic order about two decades ago. In this article we present a solution to the long-standing problem of their resummation beyond this order, and carry out the first complete next-to-leading logarithmic calculation for non-global observables. This is achieved by solving numerically the recently derived set of non-linear differential equations which describe the evolution of soft radiation in the planar, large-Nc limit. As a case study we address the calculation of the transverse energy distribution in the interjet rapidity region in e+e−→ dijet production. The calculation is performed by means of an algorithm that we formulate in the language of jet-calculus generating functionals, which also makes the resummation technique applicable to more general non-global problems, such as those that arise in hadronic collisions. We find that NLL corrections are substantial and their inclusion leads to a significant reduction of the perturbative scale uncertainties for these observables. The computer code used in the calculations is made publicly available.

The F-theorem in the melonic limit

The F-theorem states that in three dimensions the sphere free energy of a field theory must decrease between ultraviolet and infrared fixed points of the renormalization group flow, and it has been proven for unitary conformal field theories (CFTs).We consider here the long-range bosonic O(N)3 model on a spherical background, at next-to-next-to-leading order of the 1/N expansion. The model displays four large-N fixed points and we test and confirm the F-theorem holds in this case. This is non-trivial as one of the couplings is imaginary, and therefore the model is non-unitary at finite N. Despite this, several tests indicating that the large-N CFTs are in fact unitary have been performed: for instance all the OPE coefficients computed so far in the large-N limit are real, and the spectrum of bilinear operators is real and above unitarity bounds. Our result, namely that the F theorem holds at large N, can be viewed as further indication that such theories are unitary.As an added bonus, we show how conformal partial waves expansions in conformal field theory can be used to resum infinite classes of vacuum diagrams. Non-perturbatively, the jump in the value of the free energy has the interpretation of the inclusion at the ultraviolet fixed point of an extra non-normalizable contribution in the conformal partial wave expansion. This can be seen in perturbation theory as the reversal of the sign of an infinite class of diagrams due to the flow of a coupling constant.

Higgs boson production and quark scattering amplitudes at high energy through the next-to-next-to-leading power in quark mass

We study the amplitudes of the quark scattering by an external electromagnetic field and of the light quark mediated Higgs boson production via gluon fusion in the high-energy limit. The asymptotic behavior of the quark form factors is obtained in the double-logarithmic approximation to all orders in strong coupling constant through mathcal{O}left({m}_q^3right) in the small quark mass expansion and the asymptotic formula is given in a closed analytic form. In the case of the two-gluon Higgs boson form factor we obtain a complete analytic result for the three-loop mathcal{O}left({m}_q^3right) double-logarithmic term while the all-order analysis is performed in the large-Nc limit of QCD and for the abelian gauge group. An estimate of the high-order high-power light quark mass effect in the Higgs boson production and decay is given.

Two Schrödinger-like Equations for hadrons

In this talk, based on [1, 2], I argue that the holographic Schrödinger Equation of (3 +1)-dim, conformal light-front QCD and the ’t Hooft Equation of (1+1)-dim, large Nc QCD, can be complementary to each other in providing a first approximation to hadron spectroscopy. Together, the two equations play a role in hadronic physics analogous that of the ordinary Schrödinger Equation in atomic physics.

Medium evolution of a static quark-antiquark pair in the large Nc limit

We study the transitions between the different color states of a static quark-antiquark pair, singlet and octet, in a thermal medium. This is done non-perturbatively exploiting the infinite mass limit of QCD. This study is interesting because it can be used for future developments within the framework of Effective Field Theories (EFTs) and because it can be combined with other techniques, like lattice QCD or AdS/CFT, to gain non-perturbative information about the evolution of quarkonium in a medium. We also study the obtained expressions in the large Nc limit. This allows us to learn lessons that are useful to simplify phenomenological models of quarkonium in a plasma.

A brief overview of exotic hadron states

The present manuscript gives an insight into exotic hadron states which are termed tetraquarks and pentaquarks. Their existence had been postulated even before the development of the quark model and recently their detection has been pointed at different accelerator centers. Some studies consider these exotic states to be hadronic molecules while other studies disregard this theory. A lot of anomalies exists in the interpretation of the X(3872) particle which stands as a strong tetraquark candidate. Different models and approaches have been theorized to explain the structure or to calculate the physical parameters of exotic hadrons. Both, the flux tube model and large Nc limit have proven to be successful in determining the intricacies of these exotic hadron states. The partially and fully heavy exotic hadrons have generated a lot of interest for study and the chiral quark model is used to specifically study the qqqq¯Q pentaquark. A lot of theoretical and experimental intervention is required to intrinsically study the behavior and actual structure of states.

Slightly broken higher spin symmetry: general structure of correlators

We explore a class of CFT’s with higher spin currents and charges. Away from the free or N = ∞ limit the non-conservation of currents is governed by operators built out of the currents themselves, which deforms the algebra of charges by, and together with, its action on the currents. This structure is encoded in a certain A∞/L∞-algebra. Under quite general assumptions we construct invariants of the deformed higher spin symmetry, which are candidate correlation functions. In particular, we show that there is a finite number of independent structures at the n-point level. The invariants are found to have a form reminiscent of a one-loop exact theory. In the case of Chern-Simons vector models the uniqueness of the invariants implies the three-dimensional bosonization duality in the large-N limit.

Sp(4) gauge theories and beyond the standard model physics

We review numerical results for models with gauge group Sp(2N), discussing the glueball spectrum in the large-N limit, the quenched meson spectrum of Sp(4) with Dirac fermions in the fundamental and in the antisymmetric representation and the Sp(4) gauge model with two dynamical Dirac flavours. We also present preliminary results for the meson spectrum in the Sp(4) gauge theory with two fundamental and three antisymmetric Dirac flavours. The main motivation of our programme is to test whether this latter model is viable as a realisation of Higgs compositeness via the pseudo Nambu Goldstone mechanism and at the same time can provide partial top compositeness. In this respect, we report and briefly discuss preliminary results for the mass of the composite baryon made with two fundamental and one antisymmetric fermion (chimera baryon), whose physical properties are highly constrained if partial top compositeness is at work. Our investigation shows that a fully non-perturbative study of Higgs compositeness and partial top compositeness in Sp(4) is within reach with our current lattice methodology.

Thermalization of randomly coupled SYK models

We investigate the thermalization of Sachdev–Ye–Kitaev (SYK) models coupled via random interactions following quenches from the perspective of entanglement. Previous studies have shown that when a system of two SYK models coupled by random two-body terms is quenched from the thermofield double state with sufficiently low effective temperature, the Rényi entropies do not saturate to the expected thermal values in the large-N limit. Using numerical large-N methods, we first show that the Rényi entropies in a pair SYK models coupled by two-body terms can thermalize, if quenched from a state with sufficiently high effective temperature, and hence exhibit state-dependent thermalization. In contrast, SYK models coupled by single-body terms appear to always thermalize. We provide evidence that the subthermal behavior in the former system is likely a large-N artifact by repeating the quench for finite N and finding that the saturation value of the Rényi entropy extrapolates to the expected thermal value in the N → ∞ limit. Finally, as a finer grained measure of thermalization, we compute the late-time spectral form factor of the reduced density matrix after the quench. While a single SYK dot exhibits perfect agreement with random matrix theory, both the quadratically and quartically coupled SYK models exhibit slight deviations.