Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA_μ(x) = ?_μ,…, with □Λ = 0, and to the operator gauge transformation δA_μ(x) = ?_μφ(x), …, δφ(x) = α^?1?·A(x), with □φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations $\text{δA}{}_{\mathrm{\mu }}\text{(x) ～}\textcolor{red}{}{}_{\mathrm{\mu }}\text{C(x)}$, …, introduced by Becchi, Rouet and Stora, where

_μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation $\text{δA}{}_{\mathrm{\mu }}\text{(x) = (Z}{}_1\text{/Z}{}_3\text{)gA}{}_{\mathrm{\mu }}\text{(x) ×}\text{Λ}\text{(x)+?}{}_{\mathrm{\mu }}\text{Λ}\text{(x)}$, …, is never a symmetry transformation and is never finite in perturbation theory. Only for $\text{Λ}\text{(x) = (Z}{}_3\text{/Z}{}_1\text{)L}$ with L finite constants or for $\text{Λ}\text{(x) = Ω}\text{z}\text{?}{}_3\text{C(x)}$ with Ω a finite constant does it become a finite symmetry transformation, where $\text{z}\text{?}{}_3$ is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A_μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.     

Gravitational fixed points and asymptotic safety from perturbation theory

The fixed point structure of the renormalization flow in Einstein gravity and higher derivative gravity is investigated in terms of the background effective action. A refined perturbative framework is developed consisting of: use of a covariant operator regularization that keeps track of powerlike divergences, a non-minimal subtraction ansatz for the originally dimensionful couplings in combination with a ‘Wilsonian’ matching condition, and the construction of a one-loop effective action exactly gauge-independent on-shell in regularized form. Using this framework strictly positive fixed points for the dimensionless Newton constant gN${\mathrm{g}}_N$ and the cosmological constant λ   can be identified already in one-loop perturbation theory. The renormalization flow is asymptotically safe with respect to the nontrivial fixed points in both cases. In Einstein gravity a residual gauge dependence of the fixed points is unavoidable while in higher derivative gravity both the fixed point and the flow equations are universal. Along this flow spectral positivity of the Hessians can be satisfied, thereby meeting an essential condition for a well-defined Euclidean field theory setting. Dependence on O(10)$O(10)$ initial data is erased to accuracy 0.5%$0.5\text{\%}$ after O(100)$O(100)$ units of the renormalization mass scale and the flow settles on a λ(gN)$\lambda ({\mathrm{g}}_N)$ orbit.     

Comparison of the lattice Λ parameter with the continuum Λ parameter in massless QCD

The ratio of the scale parameter Λ in massless QCD defined on a lattice to the one in the continuum theory is determined by performing one-loop renormalization of the coupling constant. Our calculation method on a lattice directly relates Λ^lattice to the continuum one in the minimal subtraction scheme. The effect of incorporation of massless quarks depends on a parameter λ which is introduced to avoid trouble with fermions on a lattice. For λ=1, which is Wilson's value, the ratio previously calculated by Hasenfratz in the pure gauge theory is changed as follows:

$\text{Δ}{}_{\mathrm{\alpha =1}}{}^{\mathrm{MOM}}\text{Δ}{}^{\mathrm{lattice}}\text{=83.5}\text{for pure SU(3) gauge theory;}$

$\text{Δ}{}_{\mathrm{\alpha =1}}{}^{\mathrm{MOM}}\text{Δ}{}^{\mathrm{lattice}}\text{=105.7}\text{for QCD with 3 flavors;}$

$\text{Δ}{}_{\mathrm{\alpha =1}}{}^{\mathrm{MOM}}\text{Δ}{}^{\mathrm{lattice}}\text{=105.7}\text{117.0 for QCD with 4 flavors.}$

Critical properties of the lattice QCD will also be discussed briefly.     

Vacuum of the quantum Yang-Mills theory and magnetostatics

It is argued that since in asymptotically free Yang-Mills theories the quantum ground state is not controlled by perturbation theory, there is no a priori reason to believe that individual orbits corresponding to minima of the classical action dominate the Euclidean functional integral. To examine and classify the vacua of the quantum gauge theory, we propose an effective action in which the gauge field coupling constant g is replaced by the effective coupling $\text{g}\text{(t), t =}\text{ln}\text{[}\text{F}{}_{\mathrm{\mu \nu }}{}^{\mathrm{a}}\text{)}{}^2\text{μ}{}^4\text{]}$. The vacua of this model correspond to paramagnetism and perfect paramagnetism, for which the gauge field is F_μν^a = 0, and ferromagnetism, for which (F_μν^a)² = λ², i.e. spontaneous magnetization of the vacuum occurs. We show that there are no instanton solutions to the quantum effective action. The equations for a point classical source of color spin are solved, and we show that the field infrared energy becomes linearly divergent in the limit of spontaneous magnetization. This implies bag formation, and an electric Meissner effect confining the bag contents.     

What is the relativistic generalization of a linearly rising potential?

Recent progress in summing graphs of non-Abelian gauge theories focuses attention on certain manifestly Lorentz-invariant classical action-at-a-distance theories, whose solution allows for the construction of field-theoretic Green functions in the WKB approximation. The sum of graphs is of QED type, except that the gluon propagator is modified to incorporate the renormalization-group-invariant charge $\text{g}\text{(k)}$. The purely phenomenological choice $\text{g}{}^2\text{～ k}{}^{\mathrm{?2}}$ is equivalent non-relativistically to a linearly rising potential and yields a fully relativistic classical action-at-a-distance theory with exactly soluble circular orbits, whose Bohr quantization yields an asymptotic approximation to the poles of the Green function. One finds asymptotically linear Regge trajectories, but other phenomenological aspects are not as promising when only a linearly rising potential is used (in common with other phenomenological studies). As a redeeming feature, there do not appear to be any obvious pathologies of the sort familiar from string theories and ad hoc generalizations of linearly rising potentials.     

Fermion and Higgs masses as probes of unified theories

The effects of Yukawa couplings of order of the gauge couplings in the SU(3) × SU(2) × U(1) renormalization group equations governing the evolution of observable parameters such as $\text{m}{}_{\mathrm{<mtext>b</mtext>}}\text{m}{}_{\mathrm{\tau }}$ and the Higgs mass are studied systematically to one-loop order. These parameters are found to give useful constraints on the mass of the t quark, and of possible heavier fermion families, in theories with SU(5)-like boundary conditions at unification energies.     

Reciprocal relation in asymptotically free theories

We study consequences of the reciprocal relation between annihilation and deep inelastic scattering in asymptotically free theories. Annihilation structure functions fall with q² near ω = 1 and rise near ω = 0. Multiplicity growth lies between any power of $\text{lnq}\text{q}{}^2\text{μ}{}^2$ and any power of $\text{lnq}\text{q}{}^2\text{μ}{}^2$.     

Necessary and sufficient conditions for the existence of a lagrangian. The case of quasi-linear field equations

Necessary and sufficient conditions for the existence of the Lagrangian associated with given field equations of motion are investigated. For the quasi-linear equations A_ab^μν(x^λ, φ^c, φ_?^c)φ_μν^b + B_a(x^μ, φ^b, φ_ν^b) = 0, the complete necessary and sufficient conditions are obtained, resorting to the formalism of an exterior derivative. It is emphasized that, to find expressions of these conditions, the anti-symmetric parts of the second derivatives of a Lagrangian, $\text{R}{}_{\mathrm{\mu \nu }}{}^{\mathrm{ab}}\text{= (}\text{?}{}^2\text{L}\text{?φ}{}_{\mathrm{\mu }}{}^{\mathrm{a}}\text{?φ}{}_{\mathrm{\nu }}{}^{\mathrm{b}}\text{?}\text{?}{}^2\text{L}\text{?φ}{}_{\mathrm{\nu }}{}^{\mathrm{a}}\text{?φ}{}_{\mathrm{\mu }}{}^{\mathrm{b}}\text{)/2}$, which disappear in the field equations, take an important role. The procedure to construct the Lagrandian associated with the field equations is also presented.     

General theory of virtual heavy-particle effects in renormalizable field theories (I)

We develop a systematic method of isolating the effects of virtual heavy particles in renormalizable field theories. With a φ⁴-type field-theory model involving two real scalar fields (one with a heavy mass M, and the other light), we show in detail, that up to order $\text{1}\text{M}{}^2$ (but to all orders in renormalized coupling), effects of virtual heavy particles can be completely incorporated into pure light-particle theory via effective local vertices which involve operators of canonical dimension at most six. All the coupling strengths for such effective local interactions are of order $\text{1}\text{M}{}^2$ (the decoupling theorem) and are systematically calculable in renormalized perturbation theory. We also derive a closed set of Callan-Symanzik equations which are satisfied by these coupling strengths. Using these equations, we explicitly sum all the leading logarithms (i.e., log M ～ O(1)) which appear in the perturbative calculations of the effective coupling strengths.     

Instantons in conformal gravity

We study extrema of the general conformally invariant action:

$\text{S}{}_{\mathrm{c}}\text{= ∫}\text{1}\text{α}{}^2\text{C}{}^{\mathrm{abcd}}\text{C}{}_{\mathrm{abcd}}\text{+γR}{}^{\mathrm{abcd}}{}^1\text{R}{}_{\mathrm{abcd}}{}^1\text{+iθR}{}^{\mathrm{abcd}}\text{1R}{}_{\mathrm{abcd}}$

.We find the first examples in four dimensions of asymptotically euclidean gravitational instantons. These have arbitrary Euler number and Hirzebruch signature. Some of these instantons represent tunneling between zero-curvature vacua that are not related by small gauge transformations. Others represent tunneling between flat space and topologically non-trivial zero-energy initial data. A general formula for the one-loop determinant is derived in terms of the renormalization group invariant masses, the volume of space-time, the Euler number and the Hirzebruch signature.     

Dimensional reduction and flavor chirality

We show that phenomenologically realistic flavor-chiral Yang-Mills-Higgs theories in 4 dimensions can be derived by dimensional reduction of 10-dimensional vectorlike and gauge theories, where the extra 6 dimensions form a compact coset space with scale size r. The dimensional reduction often implies a symmetry breaking pattern like that of the electroweak theory, in which case it is natural to propose $\text{r ? G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Quantum effects then determine the short-distance behavior of the theory, including any additional symmetry breaking.     

Nonorthogonality corrections in the method of correlated basis functions

A set of normalized linearly independent basis functions φ₁, φ₂, …, φ_j, … generates matrix representatives $\text{H}$ and $\text{N}$ of the Hamiltonian operator and the identity. An orthonormal basis $\text{φ}{}_1$, $\text{φ}{}_2$, …, $\text{φ}{}_{\mathrm{j}}$, … generated by a Löwdin transformation is characterized by the distance in Hilbert space between $\text{φ}{}_{\mathrm{j}}$ and φ_j. The choice of positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ minimizes these distances and maximizes the diagonal elements of $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Again for positive definite $\text{N}$${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and a finite basis, 1 ? j ? p, the analysis yields a general theorem on Trace $\text{N}$${}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ (? p for all positive and negative integral values of n except n = ?1 and ? p for n = ?1).Sufficient conditions are determined which permit the application of the binomial theorem to the evaluation of the transform of $\text{H}$. Approximate formulas for the energy eigenvalues through third order in nondiagonal matrix elements are presented in a compact form containing characteristic nonorthogonality corrections depending on the exterior or interior location of the matrix element in the perturbation formulas.     

Features of dynamically broken gauge theories

We discuss several features of dynamical symmetry breaking in gauge theories of strong, weak and electromagnetic interactions. We speculate that in some such theories the fine structure constant calculable. A possible solution of the strong P and T violation problem in QCD by dynamical symmetry breaking is indicated. Self-energy divergences are absent in such models and we compute the finite electromagnetic self-energy of a quark in QCD. The mass hierarchy problem is examined. We find models in which the fermion-gauge boson mass ratio is $\text{M}{}_{\mathrm{<mtext>F</mtext>}}{}^2\text{M}{}_{\mathrm{<mtext>B</mtext>}}{}^2\text{～}\text{exp}$ ($\text{?1}\text{g}{}^2$), where g is a gauge boson coupling, which could account for the origin of weak interactions.     

The role of chirality in demarcating confinement from liberation

This paper suggests that confinement implies a “bifurcation” between chiral and vector-like worlds. The analysis of the Dyson equations of abelian gauge theories shows that the confining ansatz $\text{D(q}{}^2\text{) ～}\text{1}\text{q}{}^4$ is inconsistent with a chiral interaction.     

Surface critical exponents using the renormalization group ϵ-expansion

A Ginzburg-Landau-Wilson (GLW) model extended to include surface energy terms is used to discuss the surface critical behaviour of a system of interacting spins. A straightforward extension of the bulk renormalization group transformation is worked out to first order in ε = 4 ? d (where d is the number of spatial dimensions). In contrast to previous work, the coefficients of the quadratic (r) and quartic (u) terms in the GLW free energy are allowed to depend on the distance from the surface. To lowest order in ε, we show that the fixed point $\text{u}{}^1\text{(z)}$ is given by the usual bulk value. The RG recursion formula we obtain has a line of possible fixed points $\text{r}{}^1\text{(z)}$ associated with both the “special” and “ordinary” transitions. The fact that the GLW model can only describe systems with short-range interactions allows one to select the correct fixed point, in which $\text{r}{}^1\text{(z)}$ is given by its bulk value up to atomic distances from the surface. We discuss the ordinary and special transitions in a unified way. The correlation functions involving surface spins are worked out and the critical exponents η_| and η_⊥ are evaluated to O(ε). Our results agree with those of Lubensky and Rubin (ordinary transition) and Bray and Moore (special transition), who used less transparent methods of carrying out the RG transformation.     

Demythification of electroproduction local duality and precocious scaling

In an effort to develop a quantitative check of asymptotically free color-gauge theories, we analyze the logarithmic corrections to ξ-scaling coming from anomalous dimensions and coefficient functions of twist-two operators and compare with electroproduction data for 1 ? Q² ? 16 GeV². Excellent agreement is obtained using $\text{g}{}^2\text{(2}\text{GeV}\text{)}\text{4π}{}^2\text{= 0.17}$ for the effective quark-gluon coupling in the color-gauge theory. Effects of higher-twist operators are suppressed by powers of $\text{M}{}_0{}^2\text{Q}{}^2$. We use data from the resonance region to show $\text{M}{}_0\text{? 400}\text{MeV}$, in agreement with theoretical expectations. Our fit to νW₂ in the scaling region also describes the resonance region in the sense of Bloom-gilman local duality. We show that local duality is a consequence of the moment predictions obtained from the operator-product expansion in quantum chromodynamics. We resolve a paradox associated with local duality and spin-zero targets. Present measurements of $\text{R =}\text{σ}{}_{\mathrm{L}}\text{σ}{}_{\mathrm{T}}$ at large x and Q² are systematically higher than our predictions.     

Index theorems and supersymmetry in the soliton sector: (II). Magnetic monopoles in 3+1 dimensions

For supersymmetric Yang-Mills theories with Bogomolny-Prasad-Sommerfield monopoles, we use an open-space trace theorem on $\text{R}{}^3$ to calculate the O(?) correction to the monopole mass. For the N = 2 theory, the unrenormalized mass correction is non-vanishing (and divergent). To the same order, we calculate the quantum corrections to the central charges, and demonstrate explicitly that the Bogomolny bound is saturated. We also show that the mass correction for the N = 4 theory vanishes to O(?). Finally we discuss the renormalization of the mass correction for the N = 2 theory. This requires the parameters of the theory to be renormalized in the monopole background-field gauge, although there is no simple way known to do this. We exhibit calculations in standard gauges to show explicitly that they give gauge-dependent answers. Physical arguments based on the Dirac quantization condition suggest that the renormalized mass correction vanishes.     

The external field problem for QCD

In lattice gauge theory, many computations such as the strong coupling expansions, mean field theory, or the few plaquette models require the evaluation of the one-link integral in the presence of an arbitrary N × N complex matrix source (J). For SU(N) gauge theories, we express our general solution to the external field problem as an integral over the maximal abelian subgroup [U(1)]^N?1

where S₀ = 2Σ_kz_k cos(φ_k ? θ), z_j are eigenvalues of √JJ⁺, e^2iNθ=detJ/detJ⁺, and G is an appropriate jacobian determinant. Our explicit solution follows from differential Schwinger-Dyson equations cast in a separable form by using fermionic variables, and the special cases of N = 2, 3 and ∞ agree with earlier derivations.     

Poincaré gauge invariance and the dynamical role of spin in gravitational theory

We classify, according to the number of independent gauge fields, Poincaré gauge invariant theoretical frameworks of describing gravity into three categories. One of them may provide the dynamical definition of the spin tensor S and that of the energy-momentum tensor T, resulting in the response equation of matter to gravity with the gravitational field strengths, D′ and F, coupled to the former tensors

$\text{T}{}_{\mathrm{\nu }}{}^{\mathrm{\mu }}\text{;}{}_{\mathrm{\mu }}\text{=D′}{}_{\mathrm{\mu \lambda \nu }}\text{T}{}^{\mathrm{\mu \lambda }}\text{+F}{}_{\mathrm{\mu \lambda \nu \rho }}\text{S}{}^{\mathrm{\rho \mu \lambda }}$

, where the right-hand side represents spin force densities. In the absence of spin the response reduces to the conventional one of general relativity, i.e., without the spin forces. For the electromagnetic field the phase-gauge invariance requires the same conclusion as for a scalar field. For a spin $\text{1}\text{2}$ particle there is torsion, which deflects its trajectory from geodesic; an explicit expression for torsion takes a simple form of the axial vector current $\text{ψ}\text{γ}{}_5\text{γ}{}_{\mathrm{k}}\text{ψ}$.     

Strong coupling limit of gφ4 theory: General formalism and applications

A method, independently proposed by Kaiser and by us, to study the strong coupling limit of the Green functions is described and discussed. Although its quantum mechanical version exhibits unusual features, an application to the anharmonic oscillator indicates that the method is able to reproduce correctly known numerical results. In spite of difficulties in the setting up of a renormalization program for the theory, a preliminary study of the CS β function for the gφ⁴ interaction shows that $\text{β(g}{}_{\mathrm{<mtext>r</mtext>}}$ is asymptotically linear in the renormalization coupling constant. Evidence is given for the compatibility of this behaviour with the information that can be drawn from the known perturbative expansion.