Large Limit Research Articles

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.

Tensor renormalization group and the volume independence in 2D U(N) and SU(N) gauge theories

The tensor renormalization group method is a promising approach to lattice field theories, which is free from the sign problem unlike standard Monte Carlo methods. One of the remaining issues is the application to gauge theories, which is so far limited to U(1) and SU(2) gauge groups. In the case of higher rank, it becomes highly nontrivial to restrict the number of representations in the character expansion to be used in constructing the fundamental tensor. We propose a practical strategy to accomplish this and demonstrate it in 2D U(N) and SU(N) gauge theories, which are exactly solvable. Using this strategy, we obtain the singular-value spectrum of the fundamental tensor, which turns out to have a definite profile in the large-N limit. For the U(N) case, in particular, we show that the large-N behavior of the singular-value spectrum changes qualitatively at the critical coupling of the Gross-Witten-Wadia phase transition. As an interesting consequence, we find a new type of volume independence in the large-N limit of the 2D U(N) gauge theory with the θ term in the strong coupling phase, which goes beyond the Eguchi-Kawai reduction.

Quark sivers function at small x: spin-dependent odderon and the sub-eikonal evolution

We apply the formalism developed earlier [1, 2] for studying transverse momentum dependent parton distribution functions (TMDs) at small Bjorken x to construct the small-x asymptotics of the quark Sivers function. First, we explicitly construct the complete fundamental “polarized Wilson line” operator to sub-sub-eikonal order: this object can be used to study a variety of quark TMDs at small x. We then express the quark Sivers function in terms of dipole scattering amplitudes containing various components of the “polarized Wilson line” and show that the dominant (eikonal) term which contributes to the quark Sivers function at small x is the spin-dependent odderon, confirming the re- cent results of Dong, Zheng and Zhou [3]. Our conclusion is also similar to the case of the gluon Sivers function derived by Boer, Echevarria, Mulders and Zhou [4] (see also [5]). We also analyze the sub-eikonal corrections to the quark Sivers function using the constructed “polarized Wilson line” operator. We derive new small-x evolution equations re-summing double-logarithmic powers of αs ln2(1/x) with αs the strong coupling constant. We solve the corresponding novel evolution equations in the large-Nc limit, obtaining a sub-eikonal correction to the spin-dependent odderon contribution. We conclude that the quark Sivers function at small x receives contributions from two terms and is given byf1T⊥qxkT2=COxkT21x+C1kT21x0+⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {f}_{1T}^{\\perp q}\\left(x,{k}_T^2\\right)={C}_O\\left(x,{k}_T^2\\right)\\frac{1}{x}+{C}_1\\left({k}_T^2\\right){\\left(\\frac{1}{x}\\right)}^0+\\cdots $$\\end{document}with the function CO(x, {k}_T^2 ) varying slowly with x and the ellipsis denoting the subasymptotic and sub-sub-eikonal (order-x) corrections.

Subleading corrections to the S3 free energy of necklace quiver theories dual to massive IIA

We investigate the S3 free energy of mathcal{N} = 3 Chern-Simons-matter quiver gauge theories with gauge group U(N)r (r ≥ 2) where the sum of Chern-Simons levels does not vanish, beyond the leading order in the large-N limit. We take two different approaches to explore the sub-leading structures of the free energy. First we evaluate the matrix integral for the partition function in the ’t Hooft limit using a saddle point approximation. Second we use an ideal Fermi-gas model to compute the same partition function, but in the limit of fixed Chern-Simons levels. The resulting expressions for the free energy F = − log Z are then compared in the overlapping parameter regime. The Fermi-gas approach also hints at a universal frac{1}{6} log N correction to the free energy. Since the quiver gauge theories we consider are dual to massive Type IIA theory, we expect the sub-leading correction of the planar free energy in the large ’t Hooft parameter limit to match higher-derivative corrections to the tree-level holographic dual free energy, which have not yet been fully investigated.

Wilson line correlators beyond the large-Nc

We study hard 1 → 2 final-state parton splittings in the medium, and put special emphasis on calculating the Wilson line correlators that appear in these calculations. As partons go through the medium their color continuously rotates, an effect that is encapsulated in a Wilson line along their trajectory. When calculating observables, one typically has to calculate traces of two or more medium-averaged Wilson lines. These are usually dealt with in the literature by invoking the large-Nc limit, but exact calculations have been lacking in many cases. In our work, we show how correlators of multiple Wilson lines appear, and develop a method to calculate them numerically to all orders in Nc. Initially, we focus on the trace of four Wilson lines, which we develop a differential equation for. We will then generalize this calculation to a product of an arbitrary number of Wilson lines, and show how to do the exact calculation numerically, and even analytically in the large-Nc limit. Color sub-leading corrections, that are suppressed with a factor {N}_c^{-2} relative to the leading scaling, are calculated explicitly for the four-point correlator and we discuss how to extend this method to the general case. These results are relevant for high-pT jet processes and initial stage physics at the LHC.

Decomposing Gender Differences in Bankcard Credit Limits

<ns3:p>In this paper, we examine if there are gender differences in total bankcard limits by utilizing a data set that links mortgage applicant information with individual-level credit bureau data from 2006 to 2016. We document that after controlling for credit score, income, and demographic characteristics, male borrowers on average have higher total bankcard limits than female borrowers. Using a standard Kitagawa-Oaxaca-Blinder decomposition, we find that 87 percent of the gap is explained by differences in the effect of observed characteristics between male and female borrowers, while approximately 10 percent of the difference can be explained by differences in the levels of observed characteristics. Using a quantile decomposition strategy to analyze the gender gap along the entire bankcard credit limit distribution, we show that gender differences in bankcard limits favor female borrowers at smaller limits and favor male borrowers at larger limits. The primary factors that drive this gap have changed over time and vary across the distribution of credit limits.</ns3:p>

Exact expressions for n-point maximal U(1)Y-violating integrated correlators in SU(N) mathcal{N} = 4 SYM

The exact expressions for integrated maximal U(1)Y violating (MUV) n-point correlators in SU(N) mathcal{N} = 4 supersymmetric Yang-Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of N and τ = θ/(2π) + 4πi/ {g}_{YM}^2 , and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights (w, −w) where w = n − 4. The correlators satisfy Laplace-difference equations that relate the SU(N+1), SU(N) and SU(N−1) expressions and generalise the equations previously found in the w = 0 case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight (w, −w). For any fixed value of N the perturbation expansion of this correlator is found to start at order ( {g}_{YM}^2 N)w. The contributions of Yang-Mills instantons of charge k > 0 are of the form qkf(gYM), where q = e2πiτ and f(gYM) = O( {g}_{YM}^{-2w} ) when {g}_{YM}^2 ≪ 1. Anti-instanton contributions have charge k < 0 and are of the form {overline{q}}^{left|kright|}hat{f}left({g}_{YM}right) , where hat{f}left({g}_{YM}right)=Oleft({g}_{YM}^{2w}right) when {g}_{YM}^2 ≪ 1. Properties of the large-N expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of n-point free-field MUV correlators with the integrands of (n − 4)-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rôle of SL(2, ℤ)-covariance in the construction.

N-spike string in AdS3\xd7 S1 with mixed flux

Sigma model in AdS3× S3 background supported by both NS-NS and R-R fluxes is one of the most distinguished integrable models. We study a class of classical string solutions for N-spike strings moving in AdS3× S1 with angular momentum J in S1 ⊂ S5 in the presence of mixed flux. We observe that the addition of angular momentum J or winding number m results in the spikes getting rounded off and not end in cusp. The presence of flux shows no alteration to the rounding-off nature of the spikes. We also consider the large N-limit of N-spike string in AdS3× S1 in the presence of flux and show that the so-called Energy-Spin dispersion relation is analogous to the solution we get for the periodic-spike in AdS3− pp-wave ×S1 background with flux.