Irrelevant Terms Research Articles

Finiteness of quantum gravity coupled with matter in three spacetime dimensions

As it stands, quantum gravity coupled with matter in three spacetime dimensions is not finite. In this paper I show that an algorithmic procedure that makes it finite exists, under certain conditions. To achieve this result, gravity is coupled with an interacting conformal field theory C . The Newton constant and the marginal parameters of C are taken as independent couplings. The values of the other irrelevant couplings are determined iteratively in the loop and energy expansions, imposing that their beta functions vanish. The finiteness equations are solvable thanks to the following properties: the beta functions of the irrelevant couplings have a simple structure; the irrelevant terms made with the Riemann tensor can be reabsorbed by means of field redefinitions; the other irrelevant terms have, generically, non-vanishing anomalous dimensions. The perturbative expansion is governed by an effective Planck mass that takes care of the interactions in the matter sector. As an example, I study gravity coupled with Chern–Simons U(1) gauge theory with massless fermions, solve the finiteness equations and determine the four-fermion couplings to two-loop order. The construction of this paper does not immediately apply to four-dimensional quantum gravity.

Consistent irrelevant deformations of interacting conformal field theories

I show that under certain conditions it is possible to define consistent irrelevant deformations of interacting conformal field theories. The deformations are finite or have a unique running scale ("quasi-finite"). They are made of an infinite number of lagrangian terms and a finite number of independent parameters that renormalize coherently. The coefficients of the irrelevant terms are determined imposing that the beta functions of the dimensionless combinations of couplings vanish ("quasi-finiteness equations"). The expansion in powers of the energy is meaningful for energies much smaller than an effective Planck mass. Multiple deformations can be considered also. I study the general conditions to have non-trivial solutions. As an example, I construct the Pauli deformation of the IR fixed point of massless non-Abelian Yang-Mills theory with N_c colors and N_f <~ 11N_c/2 flavors and compute the couplings of the term F^3 and the four-fermion vertices. Another interesting application is the construction of finite chiral irrelevant deformations of N=2 and N=4 superconformal field theories. The results of this paper suggest that power-counting non-renormalizable theories might play a role in the description of fundamental physics.

How irrelevant operators affect the determination of fractional charge.

We show that the inclusion of irrelevant terms in the Hamiltonian describing tunneling between edge states in the fractional quantum Hall effect can lead to a variety of nonperturbative behaviors in intermediate energy regimes and, in particular, affect crucially the determination of charge through shot noise measurements. We show, for instance, that certain combinations of relevant and irrelevant terms can lead to an effective measured charge nue in the strong backscattering limit and an effective measured charge e in the weak backscattering limit, in sharp contrast with standard perturbative expectations. This provides a possible scenario to explain the experimental observations by Comforti et al. [Nature (London) 416, 515 (2002)]], which are so far not understood.

Fermions on the light front transverse lattice

We address the problems of fermions in light front QCD on a transverse lattice. We propose and numerically investigate different approaches of formulating fermions on the light front transverse lattice. In one approach we use forward and backward derivatives. There is no fermion doubling and the helicity flip term proportional to the fermion mass in the full light front QCD becomes an irrelevant term in the free field limit. In the second approach with symmetric derivative (which has been employed previously in the literature), doublers appear and their occurrence is due to the decoupling of even and odd lattice sites. We study their removal from the spectrum in two ways namely, light front staggered formulation and the Wilson fermion formulation. The numerical calculations in free field limit are carried out with both fixed and periodic boundary conditions on the transverse lattice and finite volume effects are studied. We find that an even-odd helicity flip symmetry on the light front transverse lattice is relevant for fermion doubling.

Accelerated overlap fermions

Numerical evaluation of the overlap Dirac operator is difficult since it contains the sign function $\epsilon(H_w)$ of the Hermitian Wilson-Dirac operator $H_w$ with a negative mass term. The problems are due to $H_w$ having very small eigenvalues on the equilibrium background configurations generated in current day Monte Carlo simulations. Since these are a consequence of the lattice discretisation and do not occur in the continuum version of the operator, we investigate in this paper to what extent the numerical evaluation of the overlap can be accelerated by making the Wilson-Dirac operator more continuum-like. Specifically, we study the effect of including the clover term in the Wilson-Dirac operator and smearing the link variables in the irrelevant terms. In doing so, we have obtained a factor of two speedup by moving from the Wilson action to a FLIC (Fat Link Irrelevant Clover) action as the overlap kernel.

Irrelevant interactions without composite operators: a remark on the universality of second-order phase transitions

We study the critical behaviour of symmetric ϕ44 theory including irrelevant terms of the form ϕ4+2n/Λ02n in the bare action, where Λ0 is the UV cutoff (corresponding, e.g., to the inverse lattice spacing for a spin system). The main technical tool is renormalization theory based on the flow equations of the renormalization group, which permits us to establish the required convergence statements in generality and rigour. As a consequence the effect of irrelevant terms on the critical behaviour may be studied to any order without using renormalization theory for composite operators. This is a technical simplification and seems preferable from the physical point of view. In this short paper we restrict ourselves for simplicity to the symmetry class of the Ising model, i.e. one-component ϕ44 theory. The method is general, however.

Analytically based estimation of the maximum amplitude during passage through resonance

Approximation formulae for both the maximum amplitude and the respective resonance frequencies of a linear oscillator passing resonance are derived analytically. Starting from the exact solution for run-up or run-down with constant acceleration, irrelevant terms are omitted. The analytically found results are compared to the known empirical approximation formulae.

Nonperturbative gauge fixing and perturbation theory

We compare the gauge-fixing approach proposed by Jona-Lasinio and Parrinello, and by Zwanziger (JPLZ) with the standard Fadeev-Popov procedure, and demonstrate perturbative equality of gauge-invariant quantities, up to irrelevant terms induced by the cutoff. We also show how a set of local, renormalizable Feynman rules can be constructed for the JPLZ procedure.

Real representation in chiral gauge theories on the lattice

The Weyl fermion belonging to the real representation of the gauge group provides a simple illustrative example for L\"uscher's gauge-invariant lattice formulation of chiral gauge theories. We can explicitly construct the fermion integration measure globally over the gauge-field configuration space in the arbitrary topological sector; there is no global obstruction corresponding to the Witten anomaly. It is shown that this Weyl formulation is equivalent to a lattice formulation based on the Majorana (left--right-symmetric) fermion, in which the fermion partition function is given by the Pfaffian with a definite sign, up to physically irrelevant contact terms. This observation suggests a natural relative normalization of the fermion measure in different topological sectors for the Weyl fermion belonging to the complex representation.

Extreme type-II superconductors in a magnetic field: A theory of critical fluctuations

A theory of critical fluctuations in extreme type-II superconductors subjected to a finite but weak external magnetic field is presented. It is shown that the standard Ginzburg-Landau representation of this problem can be recast, with help of a mapping, as a theory of a new ``superconductor,'' in an effective magnetic field whose overall value is zero, consisting of the original uniform field and a set of neutralizing unit fluxes attached to ${N}_{\ensuremath{\Phi}}$ fluctuating vortex lines. The long-distance behavior of this theory is governed by a phase transition line in the $(H,T)$ plane, ${T}_{\ensuremath{\Phi}}(H),$ along which the new ``superconducting'' order parameter $\ensuremath{\Phi}(\mathbf{r})$ attains long-range order. Physically, this phase transition arises through the proliferation, or ``expansion,'' of thermally generated infinite vortex loops in the background of field-induced vortex lines. Simultaneously, the field-induced vortex lines lose their effective line tension relative to the field direction. It is suggested that the critical behavior at ${T}_{\ensuremath{\Phi}}(H)$ belongs to the universality class of the anisotropic Higgs-Abelian gauge theory, with the original magnetic field playing the role of ``charge'' in this fictitious ``electrodynamics'' and with the absence of reflection symmetry along H giving rise to dangerously irrelevant terms. At zero field, $\ensuremath{\Phi}(\mathbf{r})$ and the familiar superconducting order parameter $\ensuremath{\Psi}(\mathbf{r})$ are equivalent, and the effective line tension of large loops and the helicity modulus vanish simultaneously, at ${T=T}_{c0}.$ In a finite field, however, these two forms of ``superconducting'' order are not the same and the ``superconducting'' transition is generally split into two branches: the helicity modulus typically vanishes at the vortex lattice melting line ${T}_{m}(H),$ while the line tension and associated \ensuremath{\Phi} order disappear only at ${T}_{\ensuremath{\Phi}}(H).$ We expect ${T}_{\ensuremath{\Phi}}(H)&gt;{T}_{m}(H)$ at lower fields and ${T}_{\ensuremath{\Phi}}{(H)=T}_{m}(H)$ for higher fields. Both \ensuremath{\Phi} and $\ensuremath{\Psi}$ order are present in the Abrikosov vortex lattice $[T&lt;{T}_{m}(H)]$ while both are absent in the true normal state $[T&gt;{T}_{\ensuremath{\Phi}}(H)].$ The intermediate \ensuremath{\Phi}-ordered phase, between ${T}_{m}(H)$ and ${T}_{\ensuremath{\Phi}}(H),$ contains precisely ${N}_{\ensuremath{\Phi}}$ field-induced vortices having a finite line tension relative to $\mathbf{H}$ and could be viewed as a ``line liquid'' in the long-wavelength limit. The consequences of this ``gauge theory'' scenario for the critical behavior in high-temperature and other extreme type-II superconductors are explored in detail, with particular emphasis on the questions of three-dimensional $\mathrm{XY}$ versus Landau level scaling, physical nature of the vortex ``line liquid'' and the true normal state (or vortex ``gas''), and fluctuation thermodynamics and transport. It is suggested that the empirically established ``decoupling transition'' may be associated with the loss of integrity of field-induced vortex lines as their effective line tension disappears at ${T}_{\ensuremath{\Phi}}(H).$ A ``minimal'' set of requirements for the theory of vortex lattice melting in the critical region is also proposed and discussed. The mean-field-based description of the melting transition, containing only field-induced London vortices, is shown to be in violation of such requirements.

A relevancy filter for constructive induction

Some machine-learning algorithms enable the learner to extend its vocabulary with new terms if, for a given a set of training examples, the learner's vocabulary is too restricted to solve the learning task. We propose a filter, called the Reduce algorithm, that selects potentially relevant terms from the set of constructed terms and eliminates terms that are irrelevant for the learning task. Restricting constructive induction (or predicate invention) to relevant terms allows a much larger explored space of constructed terms. The elimination of irrelevant terms is especially well-suited for learners of large time or space complexity, such as genetic algorithms and artificial neural networks. To illustrate our approach to feature construction and irrelevant feature elimination, we applied our proposed relevancy filter to the 20- and 24-train East-West Challenge problems. The experiments show that the performance of a hybrid genetic algorithm, RL-ICET (Relational Learning with ICET), improved significantly when we applied the relevancy filter while pre-processing the data set.

A gauge-fixing action for lattice gauge theories

We present a lattice gauge-fixing action S gf with the followingperties: (a) S gf is proportional to the trace of ( Σ μ ∂ μ A μ ) 2 , plus irrelevant terms of dimension six and higher; (b) S gf has a unique absolute minimum at U x , μ = I . Noting that the gauge-fixed action is not BRST invariant on the lattice, we discuss some important aspects of the phase diagram.

Universal relation between Green functions in random matrix theory

We prove that in random matrix theory there exists a universal relation between the onepoint Green function G and the connected two-point Green function G c given by N 2 G c( z, w) = ( ∂ 2/ ∂z ∂w) log[( G( z) − G( w))/( z − w) + irrelevant factorized terms]. This relation is universal in the sense that it does not depend on the probability distribution of the random matrices for a broad class of distributions, even though G is known to depend on the probability distribution in detail. The universality discussed here represents a different statement than the universality we discovered some time ago, which states that a 2 G c ( az, aw) is independent of the probability distribution, where a denotes the width of the spectrum and depends sensitively on the probability distribution. It is shown that the universality proved here also holds for the more general problem of a hamiltonian consisting of the sum of a deterministic term and a random term analyzed perturbatively by Brézin, Hikami, and Zee.

RENORMALIZATION IN TRUNCATED PARISI-SOURLAS MODEL

The Parisi-Sourlas model is truncated by ignoring irrelevant terms and introducing an N-vector field. The β-functions for the model of 4−ε dimensions are calculated up to two-loop order in the dimensional regularization scheme. By using the β-functions, the critical exponents are evaluated up to ε2-order. The critical exponents of the truncated Parisi-Sourlas model are compared with those of the ordinary N-vector model. It is found that this model roughly corresponds to the (N−2)-vector model. In fact, the ghost-like fields contribute to the critical exponents in the opposite way, being contrasted with the ordinary N-vector field.

On the role of the higgs mechanism in present electroweak precision tests

Based on the observables $\MW$, $\Gamma_l$, $\bar\sw^2(\MZ^2)$, we evaluate the parameters $\Delta x, \Delta y$ and $\varepsilon$ at one-loop level within an electroweak massive vector-boson theory, which does not employ the Higgs mechanism. The theoretical results are consistent with the experimental ones on $\Delta x$, $\Delta y$, $\varepsilon$. The theoretical prediction for $\Delta y$ coincides with the standard-model one (apart from numerically irrelevant terms which vanish for $\MH\to\infty$). Non-renormalizability only affects $\Delta x$ and $\varepsilon$, which differ from the standard-model results by the replacement $\log\MH\to\log\Lambda$ for a heavy Higgs mass, $\MH$ (where $\Lambda$ denotes an effective UV cut-off).

Hierarchical Mass Matrices in a Minimal SO(10) Grand Unification. I

We consider a minimal SO(10) unified model with horizontal Peccei-Quinn symmetry. The hierarchical structure of quark-lepton mass matrices is naturally implemented by the remnants of certain irrelevant terms. Georgi-Jarlskog relations are also realized due to the horizontal symmetry.

Universality classes for the dynamic surface critical behavior of systems with relaxational dynamics.

We investigate the problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist\char22{}one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$g0 called model ${\mathit{B}}_{\mathit{A}}$, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=0 called model ${\mathit{B}}_{\mathit{B}}$\char22{}and each static surface universality class splits up into two dynamic surface universality classes.

Lorentz invariance of effective strings

Starting from a Poincaré-invariant field theory of a real scalar field with interactions governed by a double-well potential in 2+1 dimensions, the Lorentz representation induced on the collective coordinates describing low-energy excitations about an effective string background is derived. In this representation, Lorentz transformations are given in terms of an infinite series, in powers of derivatives along the world-sheet. Transformations that act on the direction transverse to the string world-sheet involve a universal dimension - 1 term. As a consequence, Lorentz invariance holds in this theory of long effective strings due to cancellations in the action between irrelevant terms and the dimension-2 term that describes free massless scalar fields in two dimensions.

Anomalous multifractality of conductance jumps in a hierarchical percolation model

The authors have investigated the multifractal scaling of conductance jumps in a hierarchical percolation lattice, resulting from cutting the current carrying bonds in the lattice. Due to the iterative nature of the model, exact renormalization group (RG) equations are obtained and used to extract the minimum conductance jump of the lattice. They find an asymptotically analytic expression for the minimum conductance jump, Delta gmin approximately=exp(-c(log L)2) decreasing faster than any power law. They observed slow convergence to the asymptotic behaviour due to the importance of the irrelevant terms in the RG equations at low generations of the lattice. Numerical calculations are performed in order to validate the analytic results and to calculate the f- alpha spectrum to confirm left-sided multifractality as proposed by Lee and Stanley (1988).

Further investigations of the crumpling transition in dynamically triangulated random surfaces

We examine further the critical behaviour of dynamically triangulated random surfaces (DTRS) with extrinsic curvature at their second-order crumpling transition. We show that the string tension in these models may be scaling near the transition in such a way that the physical string tension is finite, unlike models containing only a Polyakov term, suggesting that one can use DTRS as a discretization of subcritical string theory. We explore the universality properties of DTRS, showing that an apparently irrelevant term can affect the phase transition. We also find that the observed phase transition persists when the surfaces are embedded in higher dimensions, contradicting the naive expectations of a saddle point expansion.