Renormalizability of φ3 theory in 6-dimensional curved spacetime

We show that φ³ theory in a general 6-dimensional curved spacetime is renormalizable at the two-loop level by demonstrating the cancellation of non-local divergences. We also calculate the full set of local two-loop counterterms. The heat kernel is used to obtain an expansion for the propagator, in the presence of the background field, in configuration space. This expansion isolates singular terms in the short-distance limit which give rise to ultraviolet divergences.     

Background field calculations in curved spacetime: (I). General formalism and application to scalar fields

The ultraviolet divergences for vacuum graphs in the presence of arbitrary background fields and euclidean metric in curved space are determined for all two- and some three-loop graphs. The method is applicable to general field theories and can be extended to higher orders but is here applied to a general scalar theory to calculate the corresponding counterterms and associated β functions. The heat kernel is used to derive an expansion for the propagator that explicitly isolates the singular terms in the short-distance limit. With dimensional regularisation the poles in ? = 4?d are determined from explicit distribution formulae for the products of the singular functions that arise. The results maintain general covariance and generalise previous results for flat space. A justification for the continuation of integral expressions, in the presence of the arbitrary background, to general dimension d is given.     

Dimensional regularisation in the presence of large background fields

The method of dimensional regularisation is introduced and discussed in detail for the case of quantum field theories with large (i.e., not infinitesimal) classical background fields. Examples considered are the φ⁴-theory on four-dimensional compact curved spaces as well as Yang-Mills gauge theories on S⁴ with an arbitrary multi-instanton background field.     

The background field method and the S-matrix

We prove that the S-matrix can be correctly obtained from the gauge-invariant effective action in the background field approach to gauge theories. In addition, we present a computation of the two-loop fermionic contributions to the Yang-Mills β-function.     

Two-loop double-pole counter lagrangian

On the basis of the background field method, extended one-loop counterterm formulae are derived, and a method to get the two-loop double-pole counter lagrangian is given. As an example, the double-pole divergences of ξ²X³ counterterms are obtained.     

Unusual initial state interactions and Kinoshita diagram identities in perturbative QCD

Unlike in QED, algebraic identities for individual single-cut Kinoshita diagrams with all soft gluons are generally violated by initial state spectator interactions in light-cone time-ordered perturbation theory. At g⁴ order, identities insuring IR cancellation for the real graphs considered by Bodwin, Brodsky and Lepage for ππ→μ⁺μ^?+X when all gluons are soft can be obtained by summing over several of the corresponding single-cut diagrams. Although this fails for the two-loop virtual's diagrams considered by BBL, summing over a larger class of possible two-loop virtual spectator interactions having two Glauber singularities does restore IR finiteness when both gluons are soft.For individual single-cut Kinoshita diagrams with an arbitrary number ladder exchanges, algebraic identities for the (C_F)^n/2 contribution from the initial state are derived.     

A modified pulsed gradient technique for measuring diffusion in the presence of large background gradients

An NMR technique for measuring the diffusion constant D in the presence of a large nonuniform background magnetic field gradient G₀ is presented. The technique uses a Carr-Purcell-Meiboom-Gill of pulse train that attenuates the effects of diffusion due to the background gradient, interspersed with an alternating pulsed field gradient sequence (APFG) that attenuates the observed echo in the presence of the known applied gradient. Calculations for the observed echo amplitude are presented that show the APFG technique eliminates contributions from the cross term between the background and applied gradients. Results of tests of the technique are presented for the measurement of D in H₂O in the presence of G₀ ∼ 160 G/cm. Also described are the results of preliminary measurements of D in LaNi₅H₆; D = (6.2 ± 0.5) × 10⁻⁸ cm²/see at 331.2 K and G₀ ∼ 2.9 kG/cm.     

Effective field theory in background field gauge

It is shown how to construct effective lagrangians to the two-loop level for non-abelian gauge theories, quantized in background field gauge. This construction is no more difficult than the analogous computation in abelian gauge theories when minimal subtraction is employed. It is argued that for three or more loops, one can no longer separate heavy or mixed light-heavy graphs in an intermediate renormalization, and the more general algorithm presented by us in an earlier work must be employed even in background field gauge.     

Explicit two-loop calculation of decoupling in unbroken gauge theories with fermions

Unbroken gauge theories containing light as well as heavy fermions are considered in the limit of the mass of the heavy fermions tending to infinity. The effective coupling constant of the decoupled low-energy theory thus obtained, has been calculated up to two-loop level using the light particle irreducible vertex function and the background field formalism. In addition, to simplify the computation, a background field gauge-fixing term has been used, because in such a gauge the effective coupling constant can be calculated from a two-point function only. Our analysis reveals that in the non-abelian theory, the simple algorithm proposed by Ovrut and Schnitzer for computing the effective coupling constant up to the two-loop level is valid only in the α = 0 or α = ?3 background field gauge. A general procedure correct for all values of α is described.     

Minimal renormalization without ϵ-expansion: Amplitude functions for O(n) symmetric systems in three dimensions below Tc

Massive field theory at fixed dimension d < 4 is combined with the minimal subtraction scheme to calculate the amplitude functions of thermodynamic quantities for the O(n) symmetric Φ⁴ model below T_c in two-loop order. Goldstone singularities arising at an intermediate stage in the calculation of O(n) symmetric quantities are shown to cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the amplitude functions in zero field for the order parameter, specific heat and helicity modulus (superfluid density) in three dimensions. We also calculate the q² part of the inverse of the wavenumber-dependent transverse susceptibility χT(q) which provides an independent check of our result for the helicity modulus. The two-loop contributions to the superfluid density and specific heat below T_c turn out to be comparable in magnitude to the one-loop contributions, indicating the necessity of higher-order calculations and Padé-Borel type resummations.     

Symmetry restoration and the background field method in gauge theories

Spontaneously broken gauge theories in a constant external electromagnetic field are shown to exhibit a first-order phase transition to a restored symmetry phase when the external field exceeds a certain critical value. The effects of fields characterized by various values of the two Lorentz invariants $\text{F}{}_1\text{=}\text{1}\text{2}\text{(B}{}^2\text{? E}{}^2\text{)}$ and F₂ = E · B are discussed. In a simple SU(2) model the critical field strength is found to be g_R²(F₁)_crit = 0.057 m_w⁴, m_w being the vector boson mass. A number of theoretical developments in the background field formalism are presented. A new gauge-fixing term, the background field R gauge, is introduced. The configuration space heat kernel method for evaluating functional determinants, extended to allow the use of dimensional regularization, is employed, and it is shown how to perform background field calculations in a gauge specified by an arbitrary parameter α. Further applications of these methods are discussed.     

Advection of passive magnetic field by the Gaussian velocity field with finite correlations in time and spatial parity violation

Using the field theoretic renormalization group technique the model of passively advected weak magnetic field by an incompressible isotropic helical turbulent flow is investigated up to the second order of the perturbation theory (two-loop approximation) in the framework of an extended Kazantsev-Kraichnan model of kinematic magnetohydrodynamics. Statistical fluctuations of the velocity field are taken in the form of a Gaussian distribution with zero mean and defined noise with finite correlations in time. The two-loop analysis of all possible scaling regimes is done and the influence of helicity on the stability of scaling regimes is discussed and shown in the plane of exponents ? ? η, where ? characterizes the energy spectrum of the velocity field in the inertial range E ∞ k ^{1 ? 2ε}, and η is related to the correlation time at the wave number k which is scaled as k ^{?2 + η}. It is shown that in non-helical case the scaling regimes of the present vector model are completely identical and have also the same properties as those obtained in the corresponding model of passively advected scalar field. Besides, it is also shown that when the turbulent environment under consideration is helical then the properties of the scaling regimes in models of passively advected scalar and vector (magnetic) fields are essentially different. The results demonstrate the importance of the presence of a symmetry breaking in a given turbulent environment for investigation of the influence of an internal tensor structure of the advected field on the inertial range scaling properties of the model under consideration and will be used in the analysis of the influence of helicity on the anomalous scaling of correlation functions of passively advected magnetic field.     

Two-loop two-point functions with masses: asymptotic expansions and Taylor series,in any dimension

In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and smallq ², ind dimensions, using identities derived from integration by parts. Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations. Padé approximants to truncations of the expansions are shown to be of great utility. As an application, we obtain the two-loop photon self-energy, for alld, and achieve highly accelerated convergence of its expansions in powers ofq ²/m ² orm ²/q ², ford=4.     

Two-loop quantum corrections to the soliton mass in two-dimensional scalar field theories

We study higher-order corrections to the soliton mass in two-dimensional scalar field theories.We show that the second quantum correction (two-loop graphs) to the soliton mass (M_S) is finite provided one orders correctly the non-commuting operators in the effective hamiltonian. That is, the vacuum sector UV counterterm suffices to eliminate the ultraviolet and infinite volume divergences of the one-soliton sector.We evaluate explicitly the finite part of the second quantum correction to M_S in the sine-Gordon model. We find that the ratio of the soliton mass to the meson mass is the same in our perturbative calculation, as in the semiclassical one by Dashen, Hasslacher and Neveu, up to two-loop contributions.     

Hot gluon matter in a constantA 0 background

We compute the effective action of finiteT SU(3) gauge theory in a constant diagonal background fieldA ₀(t,x)=B ₀ ³ T₃+B ₀ ⁸ T₈ in the general covariant background gauge up to terms of orderg ³.B ₀ ^3,8 shield the infrared singularities and the aim is to study whether the minimum of the effective action would determine their values dynamically. We find that the orderg ² term depends explicitly on the gauge fixing parameter ξ. Since the background field screens already at the tree diagram level the interactions of the six non-diagonal gluon fields they do not contribute to the plasmon-likeg ³ term. The two diagonal fieldsA ₀ ³ ,A ₀ ⁸ do, but the electric mass squared they develop will become negative if the background fields are larger than aboutT/g. Hence large background fields make the system unstable.     

Higher order contributions in finite temperature renormalization group

We apply to the O(N) λφ⁴ model the finite temperature renormalization group method, in which the temperature-dependent effective coupling constant and mass are introduced. It is pointed out that the improvement by use of the two-loop renormalization-group coefficients is important at high temperature and drastically changes the lowest order result. We also show that the two-loop renormalization-group coefficients are exact in the large-N limit.     

Dimensional regularization in the Wess-Zumino-Witten model

A method is described for the dimensional regularization and renormalization of a two-dimensional non-linear sigma model with torsion, the Wess-Zumino-Witten model. The technique solves the problems generated by the presence of the antisymmetric tensor e^uv. The two-loop calculation agrees with the non-perturbative result.     

Spin-0 and spin-1/2 particles in a constant scalar-curvature background

We study the Klein-Gordon and Dirac equations in the presence of a background metric ds²=−dt²+dx²+e^−2gx(dy²+dz²) in a semi-infinite lab (x>0). This metric has a constant scalar-curvature R=6g² and is produced by a perfect fluid with equation of state p=−ρ/3. The eigenfunctions of spin-0 and spin-1/2 particles are obtained exactly, and the quantized energy eigenvalues are compared. It is shown that both of these particles must have nonzero transverse momentum in this background. We show that there is a minimum energy E²_min=m²c⁴+g²c²?² for bosons (E_KG>E_min), while the fermions have no specific ground state (E_Dirac>mc²).     

General background field calculations with fermion fields

Background field calculations including fermion background fields are undertaken to determine one- and two-loop counterterms, and β-functions, for a general vector, scalar, spinor field gauge theory. Dimensional regularisation, and also the dimensional reduction variant, with a non-perturbative treatment of the background fields are used. In the dimensional reduction case two alternative forms are given for the results depending on the precise treatment of ε-scalars.     

A symmetry of the string background field equations

We investigate the effect of duality transformations on the geometries of a subclass of two-dimensional non-linear sigma-models. We identify the torsion which appears on dualizing such a model, initially without Wess-Zumino-Witten term, as the field strength of the gauge connection appearing in a Kałuza-Klein interpretation of the initial geometry. We show that duality preserves quantum conformal invariance at order [α′]⁰, where α′ is the string tension parameter, provided the change induced in the geometry by duality is accompanied by a shift in the dilaton field. We interpret these combined transformations as a symmetry of the order [α′]⁰ string background field equations.