A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory

The goal of this article is to provide a practical method to calculate, in a scalar theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric phase to perform high-precision calculations of the unsubtracted Green's functions at zero momentum in dimension 3, 4, and 5. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting beta reach a critical value beta_c (with up to 16 significant digits in our actual calculations). We show that the round-off errors on the magnetic susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the systematic errors (finite truncations and volume) can be controlled with an exponential precision and reduced to a level lower than the numerical errors. We justify the use of the truncation for calculations of the high-temperature expansion. We calculate the dimensionless renormalized coupling constant corresponding to the 4-point function and show that when beta -> beta_c, this quantity tends to a fixed value which can be determined accurately when D=3 (hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.Comment: Uses revtex with psfig, 31 pages including 15 figure

Dyson instability for 2D nonlinear O(N) sigma models

For lattice models with compact field integration (nonlinear sigma models over compact manifolds and gauge theories with compact groups) and satisfying some discrete symmetry, the change of sign of the bare coupling g_0^2 at zero results in a mere discontinuity in the average energy rather than the catastrophic instability occurring in theories with integration over arbitrarily large fields. This indicates that the large order of perturbative series and the non-perturbative contributions should have unexpected features. Using the large-N limit of 2-dimensional nonlinear O(N) sigma model, we discuss the complex singularities of the average energy for complex 't Hooft coupling lambda= g_0^2N. A striking difference with the usual situation is the absence of cut along the negative real axis. We show that the zeros of the partition function can only be inside a clover shape region of the complex lambda plane. We calculate the density of states and use the result to verify numerically the statement about the zeros. We propose dispersive representations of the derivatives of the average energy for an approximate expression of the discontinuity. The discontinuity is purely non-perturbative and contributions at small negative coupling in one dispersive representation are essential to guarantee that the derivatives become exponentially small when lambda -> 0^+ We discuss the implications for gauge theories.Comment: 10 pages, 10 figures uses revte

A Two-Parameter Recursion Formula For Scalar Field Theory

We present a two-parameter family of recursion formulas for scalar field theory. The first parameter is the dimension $(D)$. The second parameter ($\zeta$) allows one to continuously extrapolate between Wilson's approximate recursion formula and the recursion formula of Dyson's hierarchical model. We show numerically that at fixed $D$, the critical exponent $\gamma$ depends continuously on $\zeta$. We suggest the use of the $\zeta -$independence as a guide to construct improved recursion formulas.Comment: 7 pages, uses Revtex, one Postcript figur

The non-perturbative part of the plaquette in quenched QCD

We define the non-perturbative part of a quantity as the difference between its numerical value and the perturbative series truncated by dropping the order of minimal contribution and the higher orders. For the anharmonic oscillator, the double-well potential and the single plaquette gauge theory, the non-perturbative part can be parametrized as A (lambda)^B exp{-C/lambda} and the coefficients can be calculated analytically. For lattice QCD in the quenched approximation, the perturbative series for the average plaquette is dominated at low order by a singularity in the complex coupling plane and the asymptotic behavior can only be reached by using extrapolations of the existing series. We discuss two extrapolations that provide a consistent description of the series up to order 20-25. These extrapolations favor the idea that the non-perturbative part scales like (a/r_0)^4 with a/r_0 defined with the force method. We discuss the large uncertainties associated with this statement. We propose a parametrization of ln((a/r_0)) as the two-loop universal terms plus a constant and exponential corrections. These corrections are consistent with a_{1-loop}^2 and play an important role when beta<6. We briefly discuss the possibility of calculating them semi-classically at large beta.Comment: 13 pages, 16 figures, uses revtex, contains a new section with the uncertainties on the extrapolations, refs. adde

Evidence for Complex Subleading Exponents from the High-Temperature Expansion of the Hierarchical Ising Model

Using a renormalization group method, we calculate 800 high-temperature coefficients of the magnetic susceptibility of the hierarchical Ising model. The conventional quantities obtained from differences of ratios of coefficients show unexpected smooth oscillations with a period growing logarithmically and can be fitted assuming corrections to the scaling laws with complex exponents.Comment: 10 pages, Latex , uses revtex. 2 figures not included (hard copies available on request

New Optimization Methods for Converging Perturbative Series with a Field Cutoff

We take advantage of the fact that in lambda phi ^4 problems a large field cutoff phi_max makes perturbative series converge toward values exponentially close to the exact values, to make optimal choices of phi_max. For perturbative series terminated at even order, it is in principle possible to adjust phi_max in order to obtain the exact result. For perturbative series terminated at odd order, the error can only be minimized. It is however possible to introduce a mass shift in order to obtain the exact result. We discuss weak and strong coupling methods to determine the unknown parameters. The numerical calculations in this article have been performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We found that optimization at even order is more efficient that at odd order. We compare our methods with the linear delta-expansion (LDE) (combined with the principle of minimal sensitivity) which provides an upper envelope of for the accuracy curves of various Pade and Pade-Borel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours.Comment: 10 pages, 16 figures, uses revtex; minor typos corrected, refs. adde

Fisher's zeros as boundary of renormalization group flows in complex coupling spaces

We propose new methods to extend the renormalization group transformation to complex coupling spaces. We argue that the Fisher's zeros are located at the boundary of the complex basin of attraction of infra-red fixed points. We support this picture with numerical calculations at finite volume for two-dimensional O(N) models in the large-N limit and the hierarchical Ising model. We present numerical evidence that, as the volume increases, the Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with a Wilson action, stabilize at a distance larger than 0.15 from the real axis in the complex beta=4/g^2 plane. We discuss the implications for proofs of confinement and searches for nontrivial infra-red fixed points in models beyond the standard model.Comment: 4 pages, 3 fig

Identifying barriers to accessing information and treatment for obstetric fistula in Niamey, Niger

Objective: To identify barriers to accessing information and treatment regarding obstetric fistula (OF) unique to Niger encountered by women referred to the National Referral Fistula Center. Method: A questionnaire was administered at the National Referral Fistula Center to 29 women with OF. Qualitative and quantitative statistics were computed. Results: The average individual was 30.4 years old, illiterate and from a rural area. 76.0% had antenatal care, the average labor time was 3.04 days, and 88.0% had a physician-assisted delivery. Barriers to information included rural dwelling, lack of education, lack of understanding of cause despite contact with health care workers, lack of knowledgeable resources to seek advice from or lack of ability/interest, not given specific information about availability of treatment, and not utilizing available resources to disseminate information. Barriers to treatment included lack of information regarding condition and treatment, traditional healer utilization, inability to access adequate care for condition, delay for childbirth recovery, permission needed to seek treatment, cost, timely treatment unavailable, and lack of social support. Conclusion: Improving efficiency of getting women to the hospital at time of delivery, prompt referrals for OF, and using cell phones for disseminating information or accessing transport may benefit women with OF in Niger

Immunogenicity and safety of a live attenuated varicella vaccine in healthy Indian children aged 9 - 24 months

Objectives. To investigate the safety of live attenuated varicella vaccine (aka strain) and the optimal virus titre/ dose required for immunogenicity in healthy South African children.Design. Double-blind randomised clinical study using two different lots of varicella vaccine, each at two different titres. Subjects were randomly allocated to groups 1, 2, 3 and 4 to receive vaccine containing a mean virus titre of 104,5, 103,1, 103,9 and 102,7 PFUs per dose respectively. Clinical signs and symptoms were followed up for 42 days post-vaccination. Specific varicella antibodies were measured by an indirect immunofluorescence method in sera obtained on day 0 and day 42.Setting. City Health Clinic, Chatsworth, Durban.Participants. A total of 200 healthy 9 - 24-month-old children were vaccinated, of whom 189 (44,5%) completed the study.Main outcome measures. Pre- and post-vaccination varicella antibody levels. Adverse events following varicella vaccination..Results. The vaccine was safe and well tolerated. No local symptoms were reported. Skin reactions were specifically solicited in this study: 21 reactions were reported in 8,5% (17/200) of children. Vesicles were reported in 2 vaccines (,,;: 10 vesicles in both cases). One serious adverse event was reported: hospitalisation for bronchopneumonia on day 16 post-vaccination which resolved without sequelae. Around day 42 postvaccination (range 35 - 63 days) all the 176 initially seronegative subjects had seroconverted for varicella antibodies. Post-vaccination geometric mean titres (GMTs) were 104,1, 66,2, 69,5 and 77,0 for groups 1 - 4 respectively. Six subjects who were initially seropositive maintained or increased their titres post-vaccination; 3 of the 6 showed a booster response (a ;:;, 4-fold increase from the pre-vaccination titre).Conclusions. Varicella vaccine was found to be safe, immunogenic and well tolerated. No difference in seroconversion rates or GMTs, either between groups receiving the two vaccine lots or between groups receiving the different titres of each lot, was shown

Phases of Chiral Gauge Theories

We discuss the behavior of two non-supersymmetric chiral SU(N) gauge theories, involving fermions in the symmetric and antisymmetric two-index tensor representations respectively. In addition to global anomaly matching, we employ a recently proposed inequality constraint on the number of effective low energy (massless) degrees of freedom of a theory, based on the thermodynamic free energy. Several possible zero temperature phases are consistent with the constraints. A simple picture for the phase structure emerges if these theories choose the phase, consistent with global anomaly matching, that minimizes the massless degree of freedom count defined through the free energy. This idea suggests that confinement with the preservation of the global symmetries through the formation of massless composite fermions is in general not preferred. While our discussion is restricted mainly to bilinear condensate formation, higher dimensional condensates are considered for one case. We conclude by commenting briefly on two related supersymmetric chiral theories.Comment: 23 pages, 2 figures, ReVTeX, improved forma