Φ3 Theory Research Articles

Higher-order asymptotics and critical indexes in the [formula omitted] theory

Instanton analysis is applied to the scalar ϕ3 theory. Higher-order asymptotic expansions of the renormalization constants in the minimal subtraction scheme and 6−ϵ dimensional regularization and of the index η are found. Results of the Borel resummation for the presently known terms of the ϵ-expansion of the index η up to the order ϵ4 are presented.

Classical approximation to quantum cosmological correlations

We investigate up to which order quantum effects can be neglected in calculatingcosmological correlation functions after horizon exit. We study, as a toy model,ϕ3 theory on a de Sitter background for a massless minimally coupled scalar fieldϕ. We find that for tree level and one loop contributions in the quantum theory, a goodclassical approximation can be constructed, but for higher loop corrections this isin general not expected to be possible. The reason is that loop corrections getnon-negligible contributions from loop momenta with magnitude up to the Hubble scaleH, at which scale classical physics is not expected to be a good approximation to thequantum theory. An explicit calculation of the one loop correction to the two pointfunction supports the argument that contributions from loop momenta of scaleH are not negligible. Generalization of the arguments for the toy model to derivativeinteractions and the curvature perturbation leads to the conclusion that the leading ordersof non-Gaussian effects generated after horizon exit can be approximated quite well byclassical methods. Furthermore we compare with a theorem of Weinberg. We find thatgrowing loop corrections after horizon exit are not excluded, even in single fieldinflation.

Lorentz Invariance at Finite Temperature and Its Effect onProduction Rate and Equation of State

The effect of Lorentz invariance breaking on the production rateand the equation of state at finite temperature is investigated inthe frame of ϕ3 theory. The invariance breaking significantlychanges the off-shell degree at high temperatures.

Self consistent field method for planar φ3 theory

We continue and extend earlier work on the summation of planar graphs in φ 3 field theory, based on a local action on the world sheet. The present work employs a somewhat different version of the self consistent field (meanfield) approximation compared to the previous work on the same subject. Using this new approach, we are able to determine in general the asymptotic forms of the solutions, and in the case of one solution, even its exact form. This solution leads to formation of an unstable string, in agreement with the previous work. We also investigate and clarify questions related to Lorentz invariance and the renormalization of the solution.

An improved mean field approximation on the worldsheet for planar φ3 theory

We present an improved version of our earlier work on summing the planar graphs in φ 3 field theory. The present treatment is also based on our world sheet formalism and the mean field approximation, but it makes use of no further approximations. We derive a set of equations between the expectation values of the world sheet fields, and we investigate them in certain limits. We show that the equations can give rise to (metastable) string forming solutions.

Three-body interactions, angular momentum and black hole moduli spaces

We investigate the dynamics of a pair of (4 + 1)-dimensional black holes in the moduli approximation and with fixed angular momentum. We find that spinning black holes at small separations are described by the de Alfaro, Fubini and Furlan model. For more than two black holes, we find an explicit expression for the three-body interactions in the moduli metric by associating them with the one-loop three-point amplitude of a four-dimensional ϕ3 theory. We also investigate the dynamics of a three-black-hole system in various approximations.

NEW EQUATIONS FOR VACUUM CORRELATORS IN FIELD THEORY

Stochastic quantization is used to derive exact equations connecting an infinite set of multilocal field correlators in the φ3 theory and gluodynamics. In the latter case the obtained equations are explicitly gauge-invariant. In the bilocal approximation, corresponding to the Gaussian stochastic ensemble, a minimal finite system of equations is obtained and investigated in the lowest orders of perturbation theory. A gauge-invariant diagrammatic technique in gluodynamics is developed.

High-energy, large-momentum-transfer processes: The N-rung ladder diagram of φ3 theory

The asymptotic form of the amplitude for the N-rung ladder diagram in φ3 theory is calculated in the limit s≫‖t‖≫m2. The result is proportional to s−1‖t‖−N+1 times a homogeneous polynomial of degree 2N−2 in ln s and ln‖t‖. It is not an analytic function of s and t, although the first N+1 derivatives are continuous in the physical region. Points of nonanalyticity occur when ln‖t‖=γ ln s for γ= (1)/(2) , (1)/(3) ,..., 1/(N−2). The notion of reduced Symanzik functions is introduced. With the aid of these reduced Symanzik functions, a generating function is derived for the leading logarithms of all ladder diagrams.

PERCOLATION ON A FRACTAL WITH THE STATISTICS OF PLANAR FEYNMAN GRAPHS: EXACT SOLUTION

A new bond-percolation problem on a graph (fractal) randomly chosen from all planar Feynman graphs of zero-dimensional φ3 (or φ4) theory in the thermodynamical limit (infinite order of graphs) is solved exactly. At the percolation transition point pc the mean number of clusters per volume unit has the singularity (pc−p)4 log (pc−p) which corresponds to the critical exponent α=−2. This model is a particular example of Potts models on dynamical planar lattice1 and the result agrees with the formulae obtained in Ref. 2 by conformal field theory approach for Potts spins interacting with 2D quantum gravity.

A new technique for the evaluation of Feynman diagrams in the high energy, large momentum transfer limit

The problem of finding scattering amplitudes in the high energy, large momentum transfer limit is reconsidered. We propose a novel technique of evaluating Feynman diagrams in this regime with higher order perturbative corrections in mind. The relation to other methods is discussed and the ladder diagrams in φ3 theory, recalculated up to sixth order, serve as examples.

High-energy, large-momentum-transfer processes: Ladder diagrams in ϕ3 theory (II)

The scattering amplitude for the four-rung ladder diagram in ϕ 3 theory is evaluated at high energies and for large momentum transfers. The result takes the form of s −1| t| −3 multiplied by a homogeneous sixth-order polynomial in ln s and ln| t| is larger or less than 1 2 ln s . Thus the asymptotic formula is not analytic at ln|t| = 1 2 ln s , although the first five derivatives are continuous.

Renormalizability of φ3 theory in 6-dimensional curved spacetime

We show that φ 3 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.

Nonrenormalizability of super ?3 theory for the N=2 vector gauge supermultiplet

The super Φ3 theory in N=2 superspace is considered for the N=2 vector gauge supermultiplet. Its nonrenormalizability is demonstrated at the one loop level by using supergraph techniques.

High-energy behaviour in ϕ3 theory in six dimensions

We study the high-energy behaviour of ϕ 3 theory in six dimensions in both the Regge and deep inelastic limits, using the renormalisation group and the Bethe-Salpeter equation. We consider both the case when the theory is asymptotically free, and the case when the ultraviolet behaviour is controlled by a fixed point of the renormalisation group. In the Regge limit, we find moving Regge poles in the first case, and a fixed square-root branch point in the second. In the deep inelastic limit, we confirm the results of the operator product expansion, namely that Bjorken scaling of the moments of the structure function is broken by powers of In Q 2, and Q 2 respectively in the two cases.

Φ3 theory in six dimensions and the renormalization group

Φ 3 field theory in six space-time dimensions is used as a testing ground for the renormalization group. The consistency of the new method devised by 't Hooft is verified at the two-loop level by exhibiting certain cancellations among single pole coefficients, and also relations among double and single pole coefficients. This calculation is contrasted in efficiency with one using the Gell-Mann-Low equation in dimensionally regularized form for which a solution is obtained at the one-loop level, without neglecting mass dependence. The theory shares with asymptotically free ones the virtue of an asymptotically vanishing effective coupling constant. Indeed, the next to leading term in the beta function is also of opposite sign to the coupling constant.

Quantum field theory and the coloring problem of graphs

The φk theory is compared with the multilinear theory of scalar fields φ1, φ2, ..., φk having the same mass as that of φ. In particular, it is shown that Feynman integrals encountered in the φ3 theory are not necessarily present also in the φ1 φ2 φ3 theory, but they are if they correspond to planar Feynman graphs having no tadpole part. Furthermore, a necessary and sufficient condition for the presence of a φ3 Feynman integral in the φ1 φ22 theory is found. Those considerations are applications of graph theory, especially of the coloring problem of graphs, to Feynman graphs.

Analytic continuation of scattering amplitudes and Padé approximants

We derive a representation of the scattering amplitude in the form of a Taylor series with larger domain of convergence than the Lehmann ellipse-Mandelstam analyticity in the physical plane assumed. The coefficients of this Taylor series are Legendre series evaluated in the physical region if the amplitude is analytically continued into the region of the double spectral function - the analytic continuation on a cut being understood to be done on either side. In order to improve the rate of convergence of the Legendre series, Padé approximants for Legendre series are introduced. These Padé approximants make possible the crossing of scattering amplitudes as well, which has been checked numerically by crossing the box graph in a φ3 theory from the s-channel to the t-channel, starting from its s-channel partial wave expansion.

Regge cuts and the eikonal approximation in field theory. - I.

We delve further into the mechanisms responsible for obfuscating the true eikonal content of individual Feynman graphs. These complications originate in each case from a class of intermediate states which cancel ultimately upon summation of the relevant criss-crossed sets. All fractions of the eikonal stem fromnonleading behaviour ofnonplanar graphs. Our characterization of intermediate states leads in a natural way to a counting procedure which accounts for all the eikonal contributions of criss-crossed ladder plus Born exchange iterations in φ3 theory.

Mixing of Regge Poles and Cuts in a Field-Theory Model

The Regge poles generated by ladder diagrams in a λ φ3 theory are mixed with the moving cuts generated by a class of nonplanar graphs containing an internal ladder. Since the contribution from the cut-generating diagram is of order t−2ln mt, where t is the asymptotic variable, the t−2 behavior of the pure ladder graphs is examined and the trajectory of the Regge pole near ℓ = −2 is calculated. The Mellin transform method is used throughout. The transformed amplitude corresponding to a single cut insertion is given by a product of the form pole-cut-pole, where it is only the second Regge trajectory that mixes with the cut. The cut itself depends on the leading trajectory. This result substantiates the predictions as to form of other work based on unitarity, but differs in that the cut and pole depend on different trajectory functions. Finally, multiple insertions of the cut diagrams are shown to generate an amplitude with two moving poles on each sheet of the cut.

High-Energy Behavior at Fixed Angle for the Five-Point Function in Perturbation Theory

The validity of the d-line method used by Halliday in studying the high-energy behavior at fixed angle of the four-point function is examined in more detail. This is carried out within the context of determining the high-energy behavior at fixed angle for the five-point function. The leading asymptotic behavior of the sum of all planar five-point graphs in a φ3 theory at fixed angle is s−2.