Nonperturbative Relation Research Articles

Squeezing fNL out of the matter bispectrum with consistency relations

We show how consistency relations can be used to robustly extract the amplitude of local primordial non-Gaussianity (${f}_{\mathrm{NL}}$) from the squeezed limit of the matter bispectrum, well into the nonlinear regime. First, we derive a nonperturbative relation between primordial non-Gaussianity and the leading term in the squeezed bispectrum, revising some results present in the literature. This relation is then used to successfully measure ${f}_{\mathrm{NL}}$ from $N$-body simulations. We discuss the dependence of our results on different scale cuts and redshifts. Specifically, the analysis is strongly dependent on the choice of the smallest soft momentum, ${q}_{\mathrm{min}}$, which is the most sensitive to primordial bispectrum contributions, but is largely independent of the choice of the largest hard momentum, ${k}_{\mathrm{max}}$, due to the non-Gaussian nature of the covariance. We also show how the constraints on ${f}_{\mathrm{NL}}$ improve at higher redshift, due to a reduced off-diagonal covariance. In particular, for a simulation with ${f}_{\mathrm{NL}}=100$ and a volume of $(2.4\text{ }\text{ }\mathrm{Gpc}/h{)}^{3}$, we measure ${f}_{\mathrm{NL}}=98\ifmmode\pm\else\textpm\fi{}12$ at redshift $z=0$ and ${f}_{\mathrm{NL}}=97\ifmmode\pm\else\textpm\fi{}8$ at $z=0.97$. Finally, we compare our results with a Fisher forecast, showing that the current version of the analysis is satisfactorily close to the Fisher error. We regard this as a first step towards the realistic application of consistency relations to constrain primordial non-Gaussianity using observations.

Flowing in discrete gravity models and Ward identities: a review

Ward–Takahashi identities are nonperturbative relations between correlation functions and arising from symmetries in quantum and statistical fields theories, as Noether currents conservation for classical theories. Since their historical origin, these identities were considered to prove the exact relation between counter-terms to all order of the perturbative expansion. Recently they have been considered in relation with nonperturbative renormalization group aspects for some classes of quantum field theories namely tensorial group field theories and matrix models, both characterized by a specific non-locality in their interactions, and expected to provide discrete models for quantum gravity. In this review, we summarize the state of the art, focusing on the conceptual aspects rather than technical subtleties, and provide a unified reflection on this novel and promising way of investigation. We attached great importance to the pedagogy and the self-consistency of the presentation.

Nonperturbative matching between equal-time and lightcone quantization

We investigate the nonperturbative relation between lightcone (LC) and standard equal-time (ET) quantization in the context of λϕ4 theory in d = 2. We discuss the perturbative matching between bare parameters and the failure of its naive nonperturbative extension. We argue that they are nevertheless the same theory nonperturbatively, and that furthermore the nonperturbative map between bare parameters can be extracted from ET perturbation theory via Borel resummation of the mass gap. We test this map by using it to compare physical quantities computed using numerical Hamiltonian truncation methods in ET and LC.

Reconstructing Nonequilibrium Regimes of Quantum Many-Body Systems from the Analytical Structure of Perturbative Expansions

We propose a systematic approach to the non-equilibrium dynamics of strongly interacting many-body quantum systems, building upon the standard perturbative expansion in the Coulomb interaction. High order series are derived from the Keldysh version of determinantal diagrammatic Quantum Monte Carlo, and the reconstruction beyond the weak coupling regime of physical quantities is obtained by considering them as analytic functions of a complex-valued interaction $U$. Our advances rely on two crucial ingredients: i) a conformal change of variable, based on the approximate location of the singularities of these functions in the complex $U$-plane; ii) a Bayesian inference technique, that takes into account additional known non-perturbative relations, in order to control the amplification of noise occurring at large $U$. This general methodology is applied to the strongly correlated Anderson quantum impurity model, and is thoroughly tested both in- and out-of-equilibrium. In the situation of a finite voltage bias, our method is able to extend previous studies, by bridging with the regime of unitary conductance, and by dealing with energy offsets from particle-hole symmetry. We also confirm the existence of a voltage splitting of the impurity density of states, and find that it is tied to a non-trivial behavior of the non-equilibrium distribution function. Beyond impurity problems, our approach could be directly applied to Hubbard-like models, as well as other types of expansions.

Optical drift effects in general relativity

We consider the question of determining the optical drift effects in general relativity, i.e. the rate of change of the apparent position, redshift, Jacobi matrix, angular distance and luminosity distance of a distant object as registered by an observer in an arbitrary spacetime. We present a fully relativistic and covariant approach, in which the problem is reduced to a hierarchy of ODE's solved along the line of sight. The 4-velocities and 4-accelerations of the observer and the emitter and the geometry of the spacetime along the line of sight constitute the input data. We build on the standard relativistic geometric optics formalism and extend it to include the time derivatives of the observables. In the process we obtain two general, non-perturbative relations: the first one between the gravitational lensing, represented by the Jacobi matrix, and the apparent position drift, also called the cosmic parallax, and the second one between the apparent position drift and the redshift drift. The applications of the results include the theoretical study of the drift effects of cosmological origin (so-called real-time cosmology) in numerical or exact Universe models.

One-electron singular spectral features of the 1D Hubbard model at finite magnetic field

The momentum, electronic density, spin density, and interaction dependences of the exponents that control the (k,ω)-plane singular features of the σ=↑,↓ one-electron spectral functions of the 1D Hubbard model at finite magnetic field are studied. The usual half-filling concepts of one-electron lower Hubbard band and upper Hubbard band are defined in terms of the rotated electrons associated with the model Bethe-ansatz solution for all electronic density and spin density values and the whole finite repulsion range. Such rotated electrons are the link of the non-perturbative relation between the electrons and the pseudofermions. Our results further clarify the microscopic processes through which the pseudofermion dynamical theory accounts for the one-electron matrix elements between the ground state and excited energy eigenstates.

Multichannel0→2and1→2transition amplitudes for arbitrary spin particles in a finite volume

We present a model-independent, non-perturbative relation between finite-volume matrix elements and infinite-volume $\textbf{0}\rightarrow\textbf{2}$ and $\textbf{1}\rightarrow\textbf{2}$ transition amplitudes. Our result accommodates theories in which the final two-particle state is coupled to any number of other two-body channels, with all angular momentum states included. The derivation uses generic, fully relativistic field theory, and is exact up to exponentially suppressed corrections in the lightest particle mass times the box size. This work distinguishes itself from previous studies by accommodating particles with any intrinsic spin. To illustrate the utility of our general result, we discuss how it can be implemented for studies of $N+\mathcal{J}~\rightarrow~(N\pi,N\eta,N\eta',\Sigma K,\Lambda K)$ transitions, where $\mathcal{J}$ is a generic external current. The reduction of rotational symmetry, due to the cubic finite volume, manifests in this example through the mixing of S- and P-waves when the system has nonzero total momentum.

Hidden selection rules, M5-instantons and fluxes in F-theory

We introduce a new approach to investigate the selection rules governing the contributions of fluxed M5-instantons to the F-theory four-dimensional effective action, with emphasis on the generation of charged matter F-terms. The structure of such couplings is unraveled by exploiting the perturbative and non-perturbative homological relations, introduced in our companion paper [1], which encode the interplay between the self-dual 3-form flux on the M5-brane, the background 4-form flux and certain fibral curves. The latter are wrapped by time-like M2-branes representing matter insertions in the instanton path integral. In particular, we clarify how fluxed M5-instantons detect the presence of geometrically massive U(1)s which are responsible for ‘hidden’ selection rules. We discuss how for non-generic embeddings the M5-instanton can probe ‘locally massless’ U(1) symmetries if the rank of its Mordell-Weil group is enhanced compared to that of the bulk. As a phenomenological off-spring we propose a new type of non-perturbative corrections to Yukawa couplings which may change the rank of the Yukawa matrix. Along the way, we also gain new insights into the structure of massive U(1) gauge fluxes in the stable degeneration limit.

Functional perturbative approach to equal-time correlation functions

Equal-time correlation functions play a central role in the recent noncovariant approaches to infrared QCD in the Coulomb gauge. In this article, we explore the possibilities to derive equal-time correlation functions from a generating functional, represent them diagrammatically, and formulate nonperturbative relations similar to Dyson–Schwinger equations among them, in analogy to the covariant n-point functions. For simplicity, we set up the formalism for λϕ4 theory and derive explicit expressions for the equal-time 2- and 4-point functions.

Transition form factors and mixing of pseudoscalar mesons from anomaly sum rule

Transition form factors of pi^0, eta, eta' mesons are investigated by means of the anomaly sum rule -- an exact nonperturbative relation which follows from the dispersive representation of the axial anomaly. Considering the problem of contributions of operators originated from a non-(local) operator product expansion, we found that they are required by the available set of experimental data, including the most recent data from the Belle Collaboration (which, if taken alone, can be described without such contributions, although are compatible with them). In this approach we analyzed the experimental data on eta and eta' meson transition form factors and obtained the constraints on the decay constants and mixing parameters.

Non-perturbative gauge/gravity correspondence in $ \mathcal{N} $ = 2 theories

We derive the exact supergravity profile for the twisted scalar field emitted by a system of fractional D3 branes at a $ {\mathbb{Z}_2} $ orbifold singularity supporting $ \mathcal{N} $ = 2 quiver gauge theories with unitary groups and bifundamental matter. At the perturbative level this twisted field is “dual” to the gauge coupling but it is corrected non-perturbatively by an infinite tower of fractional D-instantons. The explicit microscopic description allows to derive the gravity profile from disk amplitudes computing the emission rate of the twisted scalar field in terms of chiral correlators in the dual gauge theory. We compute these quantum correlators using multi-instanton localization techniques and/or Seiberg-Witten analysis. Finally, we discuss a non-perturbative relation between the twisted scalar and the effective coupling of the gauge theory for some simple choices of the brane set ups.

OnR-symmetric Fixed Points and Superconformality

An important unanswered question in quantum field theory is to understand precisely under which conditions scale invariance implies invariance under the full conformal group. While the general answer in two dimensions has been known for over twenty years, a precise non-perturbative relation between scale and conformal invariance in higher dimensions has been lacking. In this note, we specialize to four dimensions and give a full quantum mechanical proof that certain unitary R-symmetric fixed points are necessarily superconformal. Among other consequences, this result implies that the infrared fixed points of N=1 supersymmetric quantum chromodynamics are superconformal.

Supersymmetry and fluctuation relations for currents in closed networks

We demonstrate supersymmetry in the counting statistics of stochastic particle currents and use it to derive exact nonperturbative relations for the statistics of currents induced by arbitrarily fast time-dependent protocols.

Liouville-Lifshitz theory in3+1dimensions

We consider a four-dimensional theory in the $z=3$ Lifshitz context, with an exponential (Liouville) potential. We determine the exact renormalized potential of the theory and derive the nonperturbative relation between the renormalized and bare couplings. In addition, we show that Lorentz symmetry is naturally generated by quantum fluctuations in the infrared regime, and conclude that the model can be relevant to high energy physics.

Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles

We study the relations between two-dimensional Yang-Mills theory on the torus, topological string theory on a Calabi-Yau threefold whose local geometry is the sum of two line bundles over the torus, and Chern-Simons theory on torus bundles. The chiral partition function of the Yang-Mills gauge theory in the large N limit is shown to coincide with the topological string amplitude computed by topological vertex techniques. We use Yang-Mills theory as an efficient tool for the computation of Gromov-Witten invariants and derive explicitly their relation with Hurwitz numbers of the torus. We calculate the Gopakumar-Vafa invariants, whose integrality gives a non-trivial confirmation of the conjectured nonperturbative relation between two-dimensional Yang-Mills theory and topological string theory. We also demonstrate how the gauge theory leads to a simple combinatorial solution for the Donaldson-Thomas theory of the Calabi-Yau background. We match the instanton representation of Yang-Mills theory on the torus with the nonabelian localization of Chern-Simons gauge theory on torus bundles over the circle. We also comment on how these results can be applied to the computation of exact degeneracies of BPS black holes in the local Calabi-Yau background.

Braided Quantum Field Theories and Their Symmetries

Braided quantum field theories, proposed by Oeckl, can provide a framework for quantum field theories that possess Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions. We study Hopf algebra symmetries and such relations in the context of braided quantum field theories. We give the four algebraic conditions among Hopf algebra symmetries and braided quantum field theories that are required for the relations to hold. As concrete examples, we apply our

An exact relation for [formula omitted] orientifold field theories with arbitrary superpotential

We discuss a non-perturbative relation for orientifold parent/daughter pairs of supersymmetric theories with an arbitrary tree-level superpotential. We show that super-Yang–Mills (SYM) theory with matter in the adjoint representation at N → ∞ , is equivalent to a SYM theory with matter in the antisymmetric representation and a related superpotential. The gauge symmetry breaking patterns match in these theories too. The moduli spaces in the limiting case of a vanishing superpotential are also discussed. Finally we argue that there is an exact mapping between the effective superpotentials of two finite- N theories belonging to an orientifold pair.

Persistence properties of a system of coagulating and annihilating random walkers.

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d<2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) approximately m(zeta/d)t(-theta), with theta=dQ+Q(Q-1/2)epsilon+O(epsilon(2)), zeta=(2Q-1)epsilon+(2Q-1)(Q-1)(1/2+AQ)epsilon(2)+O(epsilon(3)), where Q=(q-1)/q, epsilon=2-d and A=-0.006....M(t) is shown to scale as M(t) approximately t(d/2-delta), where delta=d(1-Q)+(Q-1)(Q-1/2)epsilon+O(epsilon(2)). In two dimensions, we show that P(m,t) approximately ln(t)(Q(3-2Q))ln(m)((2Q-1)(2))t(-2Q) for m<<t(2Q-1). We also derive an exact nonperturbative relation between the exponents: namely delta(Q)=theta(1-Q). The one-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.

Seiberg–Witten duality in Dijkgraaf–Vafa theory

We show that a suitable rescaling of the matrix model coupling constant makes manifest the duality group of the N=2 SYM theory with gauge group SU(2). This is done by first identifying the possible modifications of the SYM moduli preserving the monodromy group. Then we show that in matrix models there is a simple rescaling of the pair ( S D , S) which makes them dual variables with Γ(2) monodromy. We then show that, thanks to a crucial scaling property of the free energy derived perturbatively by Dijkgraaf, Gukov, Kazakov and Vafa, this redefinition corresponds to a rescaling of the free energy which in turn fixes the rescaling of the coupling constant. Next, we show that in terms of the rescaled free energy one obtains a nonperturbative relation which is the matrix model counterpart of the relation between the u-modulus and the prepotential of N=2 SYM. This suggests considering a dual formulation of the matrix model in which the expansion of the prepotential in the strong coupling region, whose QFT derivation is still unknown, should follow from perturbation theory. The investigation concerns the SU(2) gauge group and can be generalized to higher rank groups.

On the chromomagnetic expectation value μG2 and higher power corrections in heavy flavor mesons

The important parameter μG2 of the heavy quark expansion is analyzed including perturbative and power corrections. The former are found small; a nonperturbative relation is suggested which allows to control power corrections. We conclude that μG2(1GeV)=0.35+0.03−0.02GeV2. The effective coupling αs(me)(ω) governing the normalization scale evolution of μG2 is given to two loops. On the nonperturbative side, we advocate the utility of combining the heavy quark expansion with expanding around the “BPS”-type approximation for the meson wavefunction, which implies relations μπ2≈μG2, −ρLS3≈ρD3 and similar ones for the nonlocal correlators.