Browsing by Author "Stevenson, Paul M."

Now showing 1 - 7 of 7
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    Effective potential calculation for a nonlocal scalar field theory
    (2006) Hassan, Syed Asif; Stevenson, Paul M.
    In the Standard Model, particles acquire mass due to interactions with a spontaneous vacuum condensate of scalar particles. The formation of the condensate can be understood as being due to a balance between the fundamental repulsive λ&phis; 4 interaction and an induced long-range attractive interaction. In order to study the role of this induced long-range interaction, we consider a model theory in which it is included, along with the standard λ&phis; 4 term, in the starting Lagrangian. We calculate the field-theoretic effective potential of this theory, working to second order in both of the interaction coupling strengths. We then investigate the renormalization of this result, in both the perturbative and the "autonomous" frameworks.
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    'Freezing' in perturbative quantum chromodynamics: The Banks-Zaks expansion and leading-b resummations
    (1998) Caveny, Scott Andrew; Stevenson, Paul M.
    We find evidence for an infrared fixed point in perturbative QCD using two complementary approximations: The Banks-Zaks expansion and the large-b resummations. Analyzing the $e\sp+e\sp-$ annihilation ratio and the polarized Bjorken-sum-rule of deep-inelastic-scattering, we use the recent calculation of the renormalization-group $\beta$-function to four-loops in QCD to confirm that the Banks-Zaks expansion is relevant when extrapolated from sixteen down to two light quark flavors. We also search for indications of infrared 'freezing' in the Borel-renormalon-inspired 'large-b' resummations. We find the 'large-b' approximation also indicates perturbative 'freezing' of the $e\sp+e\sp-$ annihilation ratio and the polarized Bjorken-sum-rule at leading order. Finally, we study the region, $n\sb{f}$ = 2, where the 'large-$n\sb{f}$' resummation and Banks-Zaks expansion complement one another. We conclude that there is strong evidence for the existence of a perturbative infrared fixed point in the Standard Model QCD.
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    Linear and nonlinear field transformations and their application in the variational approach to nonperturbative quantum field theory
    (1992) Ibanez Meier, Rodrigo; Stevenson, Paul M.
    The present work concerns nonperturbative variational studies of the effective potential beyond the Gaussian effective potential (GEP) approximation. In the Hamiltonian formalism, we study the method of non-linear canonical transformations (NLCT) which allows one to perform variational calculations with non-Gaussian trial states, constructed by nonlinear unitary transformations acting on Gaussian states. We consider in detail a particular transformation that leads to qualitative as well as quantitative improvement over the Gaussian approximation. In particular we obtain a non-trivial correction to the Gaussian mass renormalization. For a general NLCT state, we present formulas for the expectation value of the $O(N)$-symmetric $\gamma(\phi\sp2)\sp2$ Hamiltonian, and also for the one-particle NLCT state energy. We also report on the development of a manifestly covariant formulation, based on the Euclidian path integral, to construct lower-bound approximations to $\Gamma\sb{1PI}$, the generating functional of one-particle-irreducible Green's functions. In the Gaussian approximation the formalism leads to the Gaussian effective action (GEA), as a natural variational bound to $\Gamma\sb{1PI}$. We obtain, non-trivially, the proper vertex functions at non-zero momenta, and non-zero values of the classical field. In general, the formalism allows improvement beyond the Gaussian approach, by applying nonlinear measure-preserving field transformations to the path integral. We apply this method to the $O(N)$-symmetric $\lambda(\phi\sp2)\sp2$ theory. In 4 dimensions, we consider two applications of the GEA. First, we consider the N = 1 $\lambda\phi\sp4$ theory, whose renormalized GEA seems to suggest that the theory undergoes SSB, but has noninteracting particles in its SSB phase. Second, we study the Higgs mechanism in scalar quantum electrodynamics (i.e., $O(2)$ $\lambda\phi\sp4$ coupled to a U(1) gauge field) in a general covariant gauge. In our variational scheme we can optimize the gauge parameter, leading to the Landau gauge as the optimal gauge. We derive optimization equations for the GEA and obtain the renormalized effective potential explicitly.
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    Nonperturbative studies of scalar and scalar-fermion quantum field theories at zero and finite temperature using the Gaussian effective potential
    (1988) Hajj, George Antoine; Stevenson, Paul M.
    The Gaussian effective potential (GEP), a non-perturbative approach to study quantum field theory, is applied to scalar and scalar-fermion models. We study the scalar $\phi\sp6$ field coupled to fermions through g$\sb{\rm B}\phi\overline{\psi}\psi$ or g$\sb{\rm B}\phi\sp2\overline{\psi}\psi$ in 2 and 3 space-time dimensions. In addition, we derive the finite temperature (T $>$ 0) GEP from first principles and apply it to study these models at T $>$ 0. Also the Autonomous $\lambda\phi\sp4$, coupled to fermions through a Yukawa term (g$\sb{\rm B}\phi\overline{\psi}\psi$), is examined in 4 dimensions at T $>$ 0. In all these models, in order to obtain stable theories, it is found that g$\sb{\rm B}$ must vanish as 1/log(M$\sb{\rm uv}$), 1/M$\sb{\rm uv}$ or 1/M$\sbsp{\rm uv}{2}$ in 2, 3 or 4 dimensions respectively, M$\sb{\rm uv}$ being an ultraviolet cutoff which is sent to infinity. The contribution of fermions to the GEP, however, is nonvanishing. It is also found that for the class of theories discussed, symmetry, if broken, is restored above a critical temperature. The coupling constant parameter space for each model is studied carefully, and regions where symmetry breaking occurs are determined both at zero and finite temperature.
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    Second-order corrections to the Gaussian effective potential for lambda phi(4) and other theories
    (1990) Stancu, Ion; Stevenson, Paul M.
    We formulate a systematic, nonperturbative expansion for the effective potential of $\lambda\phi\sp4$ theory. At first order it gives the Gaussian effective potential (GEP), which itself contains the 1-loop and leading-order $1\over N$ results. Here, we compute the second-order terms and hence obtain the post-Gaussian effective potential (PGEP) in 1, 2, 3, and 4 spacetime dimensions. The renormalization procedure, including the calculation of the physical mass, is discussed in detail. The results in lower dimensions agree well with the GEP when the comparison is made for the same values of the bare parameters. In 4 dimensions the divergent integrals are calculated using coordinate-space methods combined with dimensional regularization. (Difficulties with other regularizations are briefly discussed). The PGEP for the "precarious" $\lambda\phi\sp4$ theory is obtained in manifestly finite form. Remarkably, the final result takes the same mathematical form as the GEP, with only some numerical co-efficients being changed. Indeed, when parametrized in terms of the physical mass and the renormalized coupling constant, only a single coefficient is changed, from 1 to 1-1/(N + 3)$\sp2$. The "autonomous" version of the 4-dimensional theory refuses to work in this approach: one obtains indeed an autonomous-like theory, but it is unbounded below for a certain range of the parameters. The influence of fermions on scalar systems is also investigated in the post-Gaussian approximation. For the simple case of an Yukawa-type coupling with no scalar self-interaction terms the results turn out to be the same as in the Gaussian approximation.
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    The isotropic N-vector model in random magnetic fields
    (1988) Stancu, Ion; Stevenson, Paul M.
    We have investigated the dynamics of the isotropic N-vector model with long-range exchange couplings in random magnetic fields using a 1/N expansion. The leading order is exactly solved, showing the existence of a ferromagnetic phase separated from the disordered paramagnetic phase by a line. The critical behaviour of the system has been examined in the next-to-leading order of the 1/N expansion, showing that the critical exponents are by no means related to the ones of the pure system in d-2 dimensions. The ordered phase has been also investigated in the next-to-leading order, revealing a typical Goldstone behaviour of the non time-persistent part of the transverse fluctuations. For the longitudinal fluctuations, two different types of coexistence singularities emerge, one from the non time-persistent (as in the pure systems), vanishing with the temperature, and a more divergent one from the time-persistent part of the correlation.
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    'Variational' optimization in quantum field theory
    (1993) Mattingly, Alan Charles; Stevenson, Paul M.
    We examine two different techniques for studying quantum field theories in which a 'variational' optimization of parameters plays a crucial role. In the context of the O(N)-symmetric $\lambda\phi\sp4$ theory we discuss variational calculations of the effective potential that go beyond the Gaussian approximation. Trial wavefunctionals are constructed by applying a unitary operator $U = e\sp{-is\pi\sb{R}\phi\sbsp{T}{2}}$ to a Gaussian state. We calculate the expectation value of the Hamiltonian using the non-Gaussian trial states generated, and thus obtain optimization equations for the variational-parameter functions of the ansatz. At the origin, $\varphi\sb{c} = 0,$ these equations can be solved explicitly and lead to a nontrivial correction to the mass renormalization, with respect to the Gaussian case. Numerical results are obtained for the (0 + 1)-dimensional case and show a worthwhile quantitative improvement over the Gaussian approximation. We also discuss the use of optimized perturbation theory (OPT) as applied to the third-order quantum chromodynamics (QCD) corrections to $R\sb{e\sp+e\sp-}.$ The OPT method, based on the principle of minimal sensitivity, finds an effective coupling constant that remains finite down to zero energy. This allows us to apply the Poggio-Quinn-Weinberg smearing method down to energies below 1 GeV, where we find good agreement between theory and experiment. The couplant freezes to a zero-energy value of $\alpha\sb{s}/\pi = 0.26,$ which is in remarkable concordance with values obtained phenomenologically.