Naive Dimensional Analysis Research Articles

Massive higher dimensional gauge fields as messengers of supersymmetry breaking

We consider theories with one or more compact dimensions with size r > 1/M, where M is the fundamental Planck scale, with the visible and hidden sectors localized on spatially separated "3-branes." We show that a bulk U(1) gauge field spontaneously broken on the hidden-sector 3-brane is an attractive candidate for the messenger of supersymmetry breaking. In this scenario scalar mass-squared terms are proportional to U(1) charges, and therefore naturally conserve flavor. Arbitrary flavor violation at the Planck scale gives rise to exponentially suppressed flavor violation at low energies. Gaugino masses can be generated if the standard gauge fields propagate in the bulk; \mu and B\mu terms can be generated by the Giudice-Masiero or by the VEV of a singlet in the visible sector. The latter case naturally solves the SUSY CP problem. Realistic phenomenology can be obtained either if all microscopic parameters are order one in units of M, or if the theory is strongly coupled at the scale M. (For the latter case, we estimate parameters by extending "naive dimensional analysis" to higher-dimension theories with branes.) In either case, the only unexplained hierarchy is the "large" size of the extra dimensions in fundamental units, which need only be an order of magnitude. All soft masses are naturally within an order of magnitude of m_{3/2}, and trilinear scalar couplings are negligible. Squark and slepton masses can naturally unify even in the absence of grand unification.

Soft supersymmetry breaking in deformed moduli spaces, conformal theories, and Script N = 2 Yang-Mills theory

We give a self-contained discussion of recent progress in computing the non-perturbative effects of small non-holomorphic soft supersymmetry breaking, including a simple new derivation of these results based on an anomaly-free gauged U(1)_R background. We apply these results to N = 1 theories with deformed moduli spaces and conformal fixed points. In an SU(2) theory with a deformed moduli space, we completely determine the vacuum expectation values and induced soft masses. We then consider the most general soft breaking of supersymmetry in N = 2 SU(2) super-Yang-Mills theory. An N = 2 superfield spurion analysis is used to give an elementary derivation of the relation between the modulus and the prepotential in the effective theory. This analysis also allows us to determine the non-perturbative effects of all soft terms except a non-holomorphic scalar mass, away from the monopole points. We then use an N = 1 spurion analysis to determine the effects of the most general soft breaking, and also analyze the monopole points. We show that naive dimensional analysis works perfectly. Also, a soft mass for the scalar in this theory forces the theory into a free Coulomb phase.

Implications of exact SUSY gauge couplings for QCD

The phase structure of SUSY gauge theories can be very different from their nonsupersymmetric counterparts. Nonetheless, there is interesting information which might be gleaned from detailed investigation of these theories. In particular, we study the precise meaning of the strong interaction scale $\Lambda$. We ask whether one can meaningfully apply naive dimensional analysis and also ask whether the study of supersymmetric theories can shed light on the apparent discrepancy between the perturbative scale $\Lambda_{QCD}$ and the ``chiral lagrangian'' scale $\Lambda_\chi$. We show that in N=1 supersymmetric Yang Mills theory, ``naive dimensional analysis'' seems to work well, with $\Lambda_\chi$ consistently equal to the scale at which the perturbatively evolved physical coupling becomes of order $4 \pi$. We turn to N=2 theories to understand better the effect of instantons in accounting for the QCD discrepancy between scales. In N=2 supersymmetric SU(2) the instanton corrections are known to all orders from the Seiberg-Witten solution and give rise to a finite scale ratio between the scale at which the perturbatively evolved and ``nonperturbatively evolved'' couplings blow up. Correspondingly, instanton effects are important even when the associated perturbatively evolved gauge coupling only gives $\alpha$ of order 1 (rather than $4 \pi$). We compare the N=2 result to instanton-induced corrections in QCD, evaluated using lattice data and the instanton liquid model, and find a remarkably similar behavior.

Naive dimensional analysis and supersymmetry

In strongly-coupled theories with no small parameters, there are factors of 4\pi that appear when the couplings of the low-energy effective lagrangian are written in units of the effective cutoff \Lambda. These numerical factors can be explained using "naive dimensional analysis." We extend these ideas to supersymmetric theories, and show how to systematically include small parameters and couplings to weakly-interacting fields. The basic principle is that if the fundamental theory is strongly coupled, then the effective theory must also be strongly coupled at the scale \Lambda. We use our results to analyze several examples where strong supersymmetric dynamics may be relevant for phenomenology. For models that break supersymmetry through strong dynamics with no small parameters, we show that the Goldstino decay constant F is of order \La^{2} / (4\pi). We also consider theories with standard-model gauge bosons coupled directly to strong supersymmetry-breaking dynamics near the weak scale; smoothly-confining theories; and a model that breaks supersymmetry through the mechanism of a deformed moduli space.

Naive dimensional analysis and truly strong interactions

We question whether the canonical estimate of a strong coupling, \alpha \approx 4\pi, is always appropriate for dynamical symmetry breaking. Our discussion is motivated by the success of naive quark models in describing low energy QCD.

Relativistic point-coupling models as effective theories of nuclei

Recent studies have shown that concepts of effective field theory such as naturalness can be profitably applied to relativistic mean-field models of nuclei. Here the analysis by Friar, Madland, and Lynn of naturalness in a relativistic point-coupling model is extended. Fits to experimental nuclear data support naive dimensional analysis as a useful principle and imply a mean-field expansion analogous to that found for mean-field meson models.

Skyrme energy functional and naturalness

Recent studies show that successful relativistic mean-field models of nuclei are consistent with naive dimensional analysis and naturalness, as expected in low-energy effective field theories of quantum chromodynamics. The nonrelativistic Skyrme energy functional is found to have similar characteristics. {copyright} {ital 1997} {ital The American Physical Society}

Counting 4π's in strongly coupled supersymmetry

We extend the "naive dimensional analysis" arguments used in QCD for estimating the strengths of operators in chiral Lagrangians to strongly coupled supersymmetric theories. In particular, we show how to count factors of 4 pi ---an inexact science, but nevertheless a useful art when such theories are used to model real particle physics.

Naturalness in the quark-meson coupling model

The quark-meson coupling (QMC) model is examined using Georgi's “naive dimensional analysis”. We argue that the QMC model is quite natural as an effective field theory for nuclei.

Top quark interactions and the search for new physics

We give a complete set of 19 independent effective dimension 5 interactions involving the top and bottom quarks with the electroweak gauge bosons within the context of the chiral Lagrangian. The anomalous operators contribute to the leading energy power E^3 of the V_L V_L -> t\bar t or t\bar b amplitudes. We also show the top quark production rates at the LHC and the LC via V_L V_L fusion processes. If no anomalous production rate is found, then these coefficients can be bound (based on the naive dimensional analysis) to be of order 10^{-2} or 10^{-1}. This is about an order of magnitude more stringent than the bounds for the next-to-leading order bosonic operators commonly studied in V_L V_L -> V_L V_L scatterings. The effect on a CP-odd observable is also briefly discussed.

Vacuum nucleon loops and naturalness

Phenomenological studies support the applicability of naturalness and naive dimensional analysis to hadronic effective lagrangians for nuclei. However, one-baryon-loop vacuum contributions in renormalizable models give rise to unnatural coefficients, which indicates that the quantum vacuum is not described adequately. The effective lagrangian framework accommodates a more general characterization of vacuum contributions without reference to a Dirac sea of nucleons.

A chiral effective lagrangian for nuclei

An effective hadronic lagrangian consistent with the symmetries of quantum chromodynamics and intended for applications to finite-density systems is constructed. The degrees of freedom are (valence) nucleons, pions and the low-lying non-Goldstone bosons, which account for the intermediate-range nucleon-nucleon interactions and conveniently describe the nonvanishing expectation values of nucleon bilinears. Chiral symmetry is realized nonlinearly, with a light scalar meson included as a chiral singlet to describe the mid-range nucleon-nucleon attraction. The low-energy electromagnetic structure of the nucleon is described within the theory using vector-meson dominance, so that external form factors are not needed. The effective lagrangian is expanded in powers of the fields and their derivatives, with the terms organized using Georgi's “naive dimensional analysis”. Results are presented for finite nuclei and nuclear matter at one-baryon-loop order, using the single-nucleon structure determined within the model. Parameters obtained from fits to nuclear properties show that naive dimensional analysis is a useful principle and that a truncation of the effective lagrangian at the first few powers of the fields and their derivatives is justified.

Generalized dimensional analysis

I give a simple qualitative derivation and interpretation of a generalization of so-called naive dimensional analysis, a rule for estimating the sizes of terms in an effective theory below the scale of chiral symmetry breaking induced by a strong gauge interaction. The generalization includes the well known N c dependence of ƒ π as a special case but I suggest that it also applies to situation very unlike QCD.

Electroweak corrections in technicolor reconsidered

Radiative corrections to electroweak parameters in technicolor theories may be evaluated by one of two techniques: either one estimates spectral function integrals using scaled QCD data, or one uses naive dimensional analysis with a chiral lagrangian. The former yields corrections to electroweak parameters proportional to the number of flavors and the number of colors, while the latter is proportional to the number of flavors squared and is independent of the number of colors. We attempt to resolve this apparent contradiction by showing that the spectrum of technicolor one obtains by scaling QCD data to high energies is unlikely to resemble that of an actual technicolor theory. The resonances are likely to be much lighter than naively supposed and the radiative corrections to electroweak parameters may by much larger. We also argue that much less is known about the spectrum and the radiative corrections in technicolor than was previously believed.

Dispersion, convection, and reaction in porous media

The problem of transport of a reactive solute in a porous medium by convection and diffusion is studied for the case in which the solute particles undergo a first-order chemical reaction on the surface of the bed. Assuming that the geometry is periodic, the method of homogenization is applied, showing explicitly that the effective equation is given by a Kramers–Moyal expansion, i.e., a partial differential equation of infinite order in which the nth term is the product of the nth gradient of the mean concentration by an nth-order constant tensor. The effective values of reactivity, solute velocity, diffusivity, and of all the tensorial coefficients in the expansion are independent of the initial solute distribution and are expressed in terms of Peclet’s and Damkohler’s numbers, Pe=aV/D and Da=ak/D, respectively, where a is the cell size, V is the solvent mean velocity, D is the solute molecular diffusivity, and k is the surface reactivity, showing that they are independent of the initial solute distribution. Since the ratio between two successive terms in the effective equation equals the small ratio ε between the micro- and macrolength scales, truncating the expansion after the nth term allows us to find the effective concentration up to O(εn) terms. The impact of this fact is exemplified, showing that in the case of a solute flowing in a pipe with small Damkohler number Da, the effective concentration can be determined up to O(Da) terms only if the effective equation includes the skewness term. When Pe and Da are either small or large, after determining a two-parameter expansion of the solution, it is shown that the ratios between the diffusion, convection, and reaction macroscopic characteristic time scales cannot always be inferred through a naive dimensional analysis at the microscale. For example, when Da≫1 we find that the effective reaction rate tends to a constant value, independent of Da. When Pe≫1, Taylor-like dispersion, proportional to Pe2, is obtained when the mean flow is perpendicular to any vector of the reciprocal lattice. If this condition is not satisfied, the result strongly depends on the magnitude of the volume fraction of the bed particles Φ. If Pe−3≪Φ≪1, then the main mechanism causing dispersion is convection alone and the effective diffusivity is proportional to Pe; on the contrary, when Φ≪Pe−3, the effective diffusivity tends to a constant value independent of Pe.

ON FERMION NUMBER VIOLATION AT HIGH ENERGIES

By considering the back reaction of the scattered particles on instantons, the explicit high energy behavior of fermion number violating amplitudes in the standard electroweak model is obtained to the leading order in the coupling constant. The amplitudes decay faster than what is expected from naive dimensional analysis due to the renormalization group flow of the coupling constant and this is consistent with unitarity. Also, it is shown that the constrained instanton approach is valid at low energies, while the high energy behavior is governed by the back reaction of the external particles.

The theory of asymptotic behavior

The Green’s functions of renormalizable quantum field theory are shown to violate, in general, Euler’s theorem on homogeneous functions, that is to say, to violate naive dimensional analysis. The respective violations are established by explicit calculation with Feynman diagrams. These violations when incorporated into the renormalization group, then provide the basis for an entirely new approach to asymptotic behavior in renormalizable field theory. Specifically, the violations add new delta-function sources to the usual partial differential equations of the group when these equations are written in terms of the external momenta of the respective Green’s functions. The effect of these sources is illustrated by studying the real part, Re Γ(6)(λp), of the six-point 1PI vertex of the massless scalar field with quartic self-coupling—the simplest of renormalizable situations. Here, Γp is symbolic for the six-momenta of Γ(6). Briefly, it is found that the usual theory of characteristics is unable to satisfy the boundary condition attendant to the respective dimensional-analysis-violating sources. Thus, the method of characteristics is completely abandoned in favor of the method of separation of variables. A complete solution which satisfies the inhomogeneous group equation and all boundary conditions is then explicitly constructed. This solution has the following interesting properties: 1) It possesses Laurent expansions in the scale λ of its momentum arguments for all real values of λ2 except λ2=0, where it has a delta-function singularity in λ2 in addition to its Laurent expansions for λ2=0+ and λ2=0−. For |λ2|→∞ and |λ2|→0, the solution’s leading term in its respective Laurent series is proportional to λ−2, so that it behaves canonically in both limits! 2) The limits λ2→0+ and λ2→0− of λ2 RE Γ(6) are both nonzero and unequal. Further, on account of the delta-function singularity at λ2=0, the value of the solution at λ2=0 is not simply related to the value of either of these limits! Apparent divergences in the coefficients in the respective Laurent expansions are removed by the use of Heisenberg’s uncertainty principle. In this way, the new approach would appear to be operationally established.

Violations of dimensional analysis in renormalized perturbation theory

The Green's functions of renormalizable quantum field theory are shown to violate, in general, Euler's theorem on homogeneous functions, that is to say, to violate naive dimensional analysis. The respective violations are established by explicit calculation with Feynman diagrams. These violations, when incorporated into the renormalization group, may provide the basis for an entirely new approach to asymptotic behavior in renormalizable field theory.