V. Elias, 2 R.B. Mann, 3 D.G.C. McKeon, and T.G. Steele Perimeter Institute for Theoretical Physics, 35 King Street North, Waterloo, ON, N2J 2W9, Canada Department of Applied Mathematics, The University of Western Ontario, London, ON, N6A 5B7, Canada Department of Physics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada Department of Physics & Engineering Physics, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada

Asymptotic Padé-approximant methods are utilized to estimate the O(α 5 s) contribution to the H → gg rate and the O(α 4 s) contribution to the H → b ¯ b rate. The former process is of particular interest because of the slow convergence evident from the three known terms of its QCD series, which begins with an O(α 2 s) leading-order term. The O(α 5 s) contribution to the H → gg rate is expressed as a degree-3 polynomial in L ≡ ln(µ 2 /m 2 t (µ)). We find that asymptotic Padé-approximant predictions for the coefficients of L, L 2 , and L 3 are respectively within 1%, 2%, and 7% of true values extracted via renormalization-group methods. Upon including the full set of next-order coefficients, the H → gg rate is found to be virtually scale-independent over the 0.3 MH < ∼ µ < ∼ Mt range of the renormalization scale-parameter µ. We conclude by discussing the small O(α 4 s) contribution to the H → b ¯ b rate, which is obtained from a prior asymptotic Padé-approximant estimate of the O(α 4 s) contribution to the quark-antiquark scalar-current correlation function.

Two approaches to renormalization-group improvement are examined: the substitution of the solutions of running couplings, masses and fields into perturbatively computed quantities is compared with the systematic sum of all the leading log (LL), next-to-leading log (NLL) etc. contributions to radiatively corrected processes, with n-loop expressions for the running quantities being responsible for summing N n-1 LL contributions. A detailed comparison of these procedures is made in the context of the effective potential V in the 4-dimensional O(4) massless λϕ 4 model, showing the distinction between these procedures at two-loop order when considering the NLL contributions to the effective potential V.

We consider in detail the analytic behaviour of the non-interacting massless scalar field two-point function in H. S. Snyder's discretized non-commuting spacetime. The propagator we find is purely real on the Euclidean side of the complex p(2) plane and goes like 1/p(2) as p(2) -> 0 from either the Euclidean or Minkowski side. The real part of the propagator goes smoothly to zero as p(2) increases to the discretization scale 1/a(2) and remains zero for p(2) > 1 a(2). This behaviour is consistent with the termination of single-particle propagation on the ultraviolet side of the discretization scale. The imaginary part of the propagator, consistent with a multiparticle-state spectral function branch discontinuity, is finite and continuous on the Minkowski side, slowly falling to zero when 1/a(2) < p(2) < infinity. The multi-particle aspect of this spectral function within the Kallen-Lehmann representation of the propagator leads to the interpretation that the propagation of free-fields in a quantized spacetime is analogous to propagation of interacting fields in a continuous spacetime.

We consider dominant three-, four- and five-loop contributions to λ, the quartic scalar coupling-constant's β-function in the Standard Model. We find that these terms accelerate the evolution of λ to nonperturbative values, thereby lowering the unification bound for which scalar-couplings are still perturbative. We also find that these higher order contributions imply a substantial lowering of λ itself before the anticipated onset of nonperturbative physics in the Higgs sector.

The top-quark Yukawa coupling is too large to permit radiative electroweak symmetry breaking to occur for small values of y, the Higgs self-coupling, to leading-logarithm order. However, a large y solution leading to a viable Higgs mass of approximately 220 GeV does exist, and differs from conventional symmetry breaking by an approximately five-fold enhancement of the Higgs self-coupling. This scenario for radiative symmetry breaking is reviewed, and the order-by-order perturbative stability of this scenario is studied within the scalar field theory projection of the standard model in which the Higgs self-coupling y represents the dominant standard-model coupling.

The perturbative β-function is known exactly in a number of supersymmetric theories and in the ‘t Hooft renormalization scheme in the [Formula: see text] model. It is shown how this allows one to compute the effective action exactly for certain background field configurations and to relate bare and renormalized couplings. The relationship between the minimal subtraction scheme and the supersymmetry subtraction scheme in N = 1 super Yang–Mills theory is discussed.PACS No.: 11.10Z

We demonstrate to two-loop order that an intermediate symmetrically embedded Pati–Salam SU(2)L×SU(2)R×SU(4) level of symmetry is all that is necessary to accommodate empirical values of α(Mz), αs(Mz) and sin2θw(Mz) within a grand unification context but with a high (1014 GeV) intermediate mass scale and with a concomitant higher GUT scale.

Accurate determinations of the (MS) over bar b-quark mass m(b)(m(b)) from sigma(e(+)e(-) -> hadrons) experimental data currently contain three comparable sources of uncertainty; the experimental uncertainty from moments of this cross-section, the uncertainty associated with alpha(s)(M-z), and the theoretical uncertainty associated with the renormalization scale. Through resummation of all logarithmic terms explicitly determined in the perturbative series by the renormalization-group (RG) equation, it is shown that the renormalization-scale dependence is virtually eliminated as a source of theoretical uncertainty in mb(mb). This resummation also reduces the estimated effect of higher-loop perturbative contributions, further reducing the theoretical uncertainties in mb(mb). Furthermore, such resummation techniques improve the agreement between the values of the (MS) over bar b-quark mass extracted from the various moments of R(s) = sigma(e(+)e(-) -> hadrons)/sigma(pt) [sigma(pt) = 4 pi alpha(2)/(3s)], obviating the need to choose an optimum moment for determining mb(mb). Based on this analysis, the resulting value of the b-mass is m(b)(m(b)) = 4.207 GeV +/- 40 MeV, where the dominant uncertainty now arises from the experimental moments. Resummation techniques are also shown to reduce renormalization-scale dependence in the relation between b-quark (MS) over bar and pole mass and in the relation between the pole and 1S mass.

We demonstrate the stability under subsequent-to-leading logarithm corrections of the quartic scalar-field coupling constant λ and the running Higgs boson mass obtained from the (initially massless) effective potential for radiatively broken electroweak symmetry in the single-Higgs-doublet Standard Model. Such subsequent-to-leading logarithm contributions are systematically extracted from the renormalization group equation considered beyond one-loop order. We show λ to be the dominant coupling constant of the effective potential for the radiatively broken case of electroweak symmetry. We demonstrate the stability of λ and the running Higgs boson mass through five orders of successively subleading logarithmic corrections to the scalar-field-theory projection of the effective potential for which all coupling constants except the dominant coupling constant λ are disregarded. We present a full next-to-leading logarithm potential in the three dominant Standard Model coupling constants (t-quark-Yukawa, αs, and λ) from these coupling constants' contribution to two loop β- and γ-functions. Finally, we demonstrate the manifest order-by-order stability of the physical Higgs boson mass in the 220–231GeV range. In particular, we obtain a 231GeV physical Higgs boson mass inclusive of the t-quark-Yukawa and αs coupling constants to next-to-leading logarithm order, and inclusive of the smaller SU(2)×U(1) gauge coupling constants to leading logarithm order.

The renormalization group (RG) is known to provide information about radiative corrections beyond the order in perturbation theory to which one has calculated explicitly. We first demonstrate the effect of the renormalization scheme used on these higher order effects determined by the RG. Particular attention is paid to the relationship between bare and renormalized quantities. Application of the method of characteristics to the RG equation to determine higher order effects is discussed and used to examine the free energy in thermal field theory, the relationship between the bare and renormalized coupling, and the effective potential in massless scalar electrodynamics.

The Wilkinson Microwave Anisotropy Probe microwave background data suggest that the primordial spectrum of scalar curvature fluctuations is suppressed at small wavenumbers. We propose a UV/IR mixing effect in small-field inflationary models that can explain the observable deviation in WMAP data from the concordance model. Specifically, in inflationary models where the inflaton couples to an asymptotically free gauge theory, the radiative corrections to the effective inflaton potential can be anomalously large. This occurs for small values of the inflaton field which are of the order of the gauge theory strong coupling scale. Radiative corrections cause the inflaton potential to blow up at small values of the inflaton field. As a result, these corrections can violate the slow-roll condition at the initial stage of the inflation and suppress the production of scalar density perturbations.

The top-quark Yukawa coupling is too large to permit radiative electroweak symmetry breaking to occur for small values of y, the Higgs self-coupling, to leading-logarithm order. However, a large y solution leading to a viable Higgs mass of approximately 220 GeV does exist, and differs from conventional symmetry breaking by an approximately five-fold enhancement of the Higgs self-coupling. This scenario for radiative symmetry breaking is reviewed, and the order-by-order perturbative stability of this scenario is studied within the scalar field theory projection of the standard model in which the Higgs self-coupling y represents the dominant standard-model coupling. Conventional electroweak (EW) symmetry breaking requires the presence of a Higgs scalar-field quadratic term in the Lagrangian. Such a mass term is unnatural if SU(2)×U(1) EW gauge theory is embedded within a grand-unified theory (GUT), since fine-tuning is needed to cancel the unification-scale perturbative corrections generated by this mass term [1] to maintain a Higgs mass empirically bounded not too far from the EW vacuum expectation value (VEV) scale 〈φ〉 = v = 246.2 GeV [2]. This fine-tuning problem can be circumvented if the embedding unified theory has a symmetry (e.g. conformal symmetry) which protects these quadratic terms from GUT-scale corrections. Radiative EW symmetry breaking provides such a scenario. Quadratic scalar mass terms are absent, and in the seminal work of Coleman & Weinberg, it is demonstrated that spontaneous symmetry breaking (i.e. the generation of a VEV) occurs via radiative (perturbative) corrections to the conformally-invariant theory [3]. Unlike conventional symmetry breaking where the Higgs mass is an unconstrained parameter, the radiative symmetry breaking mechanism is a self-consistent approach which results in a prediction of both the Higgs mass and its four-point self-coupling after imposition of the external EW scale for the VEV 〈φ〉 = v = 246.2 GeV . In the absence of large Yukawa couplings (i.e. Yukawa couplings are dominated by EW gauge couplings), a justifiable assumption at the time of Coleman & Weinberg’s work, the radiative symmetry breaking scenario contains a small-λ solution leading to an O(10 GeV) Higgs mass [3], long since ruled out via direct experimental searches. However, the top quark Yukawa coupling is large enough to destabilize this small-λ solution. The Coleman-Weinberg radiative mechanism has thus been revisited in the context of the large top-quark Yukawa coupling, revealing the persistence of a large-λ radiative scenario resulting in a 218 GeV Higgs mass for the minimal (single-Higgs-doublet) standard model [4, 5]. Consider the one-loop effective potential Veff = π φS for the Higgs sector, which must satisfy the renormalization group (RG) equation

For the Higgs boson mass of similar to 220 GeV expected to arise from radiative electroweak symmetry breaking, we find the same lowest-order expressions as would be obtained from conventional electroweak symmetry breaking, given the same Higgs boson mass, for Higgs-Goldstone sector scattering processes identified with WL+WL- -> WL+WL- WL+WL- -> Z(L)Z(L), as well as for Higgs boson decay widths H -> WL+WL-, H -> Z(L)Z(L). The radiatively broken case, however, leads to an order of magnitude enhancement over lowest-order conventional symmetry breaking for scattering processes WL+WL+ -> HH, Z(L)Z(L) -> HH, as well as a factor of similar to 30 enhancement for HH -> HH.

The effective potential for radiatively broken electroweak symmetry in the single-Higgs-doublet standard model is explored to four sequentially subleading logarithm-summation levels (5-loops) in the dominant Higgs self-interaction couplant lambda. We augment these results with all contributing leading logarithms in the remaining large but subdominant standard model couplants [t-quark, QCD and SU(2)circle times U(1) gauge couplants] as well as next to leading-logarithm contributions from the largest of these, the t-quark and QCD couplants. Order-by-order stability is demonstrated for earlier leading-logarithm predictions of an O (220 GeV) Higgs boson mass in conjunction with fivefold enhancement of the value for lambda over that anticipated from conventional spontaneous symmetry breaking.

1 Abstract The perturbative β-function is known exactly in a number of supersymmetric theories and in the 't Hooft renormalization scheme in the φ 4 4 model. It is shown how this allows one to compute the effective action exactly for certain background field configurations and to relate bare and renormalized couplings. The relationship between the MS and SUSY subtraction schemes in N = 1 super Yang-Mills theory is discussed. 2 Introduction In a number of instances, the perturbative renormalization group β-function is known exactly. In N = 4 supersymmetric Yang-Mills (SYM) theory, an β-function vanishes [1,2]. In N = 2 SYM theory, the β-function is exact at one-loop order when minimal subtraction (MS) is used [3,4]. In N = 1 SYM theory, the all-orders expression for the β-function can be determined either through instanton calculus [5] or by considering the multiplet structure of anomalies [6,7], though such an expression differs from the perburbative result derived using MS [8,9]. Although models such as the φ 4 4 scalar theory and Yang-Mills (YM) theory have all-orders contributions to the β-function in the MS scheme, one can nevertheless perform in principle a finite renormalization at each order of perturbation theory so as to have contributions to the β-function vanish beyond two-loop order and to have anomalous dimensions as entirely one-loop effects [10,11]. In this paper we demonstrate how knowledge of the full β-function can be used to extract information about the effective action. In the first instance, the effective Lagrangian in a background U(1) gauge field is determined for N = 1 and N = 2 SYM theories. In making this determination, we exploit the fact that the trace of the energy-momentum tensor θ µν is proportional to the β-function [12-15]. In ref. [16] this proportionality is used in spinor and scalar QED to determine the β-function to two-loop order from the two-loop effective action computed in the presence of a self-dual background electromagnetic field. Our approach here is to employ a known β-function to determine the effective action in the presence of a background vector field that gives rise to θ µ µ. By having restricted the background field to being a vector field and not having included any contribution from background spinor fields, SUSY invariance is lost. This sort of effective action has been considered in (46, 47) where the Euler-Heisenberg effective action in N = 4 SYM theory has …

A Comment on the Letter by P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Phys. Rev. Lett. 88, 012001 (2001).Received 24 January 2005DOI:https://doi.org/10.1103/PhysRevLett.95.099101©2005 American Physical Society

In the absence of a tree-level scalar-field mass, renormalization-group (RG) methods permit the explicit summation of leading-logarithm contributions to all orders of the perturbative series for the effective-potential functions utilized in radiative symmetry breaking. For scalar-field electrodynamics, such a summation of leading logarithm contributions leads to upper bounds on the magnitudes of both gauge and scalar-field coupling constants, and suggests the possibility of an additional phase of spontaneous symmetry breaking characterized by a scalar-field mass comparable to that of the theory's gauge boson. For radiatively-broken electroweak symmetry, the all-orders summation of leading logarithm terms involving the dominant three couplings (quartic scalar-field, t-quark Yukawa, and QCD) contributing to standard-model radiative corrections leads to an RG-improved potential characterized by a 216 GeV Higgs boson mass. Upon incorporation of electroweak gauge couplants we find that the predicted Higgs mass increases to 224 GeV The potential is also characterized by a quartic scalar-field coupling over five times larger than that anticipated for an equivalent Higgs mass obtained via conventional spontaneous symmetry breaking, leading to a concomitant enhancement of processes (such as W+W- --> ZZ) sensitive to this coupling. Moreover, if the QCD coupling constant is taken to be sufficiently strong, the tree potential's local minimum at phi = 0 is shown to be restored for the summation of leading logarithm corrections. Thus if QCD exhibits a two-phase structure similar to that of N = 1 supersymmetric Yang-Mills theory, the weaker asymptotically-free phase of QCD may be selected by the large logarithm behaviour of the RG-improved effective potential for radiatively broken electroweak symmetry. (C) 2003 Elsevier B.V. All rights reserved.

Effective actions are invariant under changes in the renormalization scale mu. The renormalization group is seen to yield differential equations whose solutions are closed form expressions for successive sums of subleading logarithmic contributions to the effective action. This procedure can be applied to any order, provided the coefficient of the zeroth order logarithm has been previously calculated at that order in perturbation theory. We demonstrate this using the phi(3)(6) model.