Unique determination of the effective potential in terms of renormalization group functions

It is first demonstrated how the effective potential Veff in a self-interacting scalar theory can be computed using operator regularization. We examine [Formula: see text] and [Formula: see text] theories, recovering the usual results in the former case and showing how Veff is a power series in the square root of the coupling λ in the latter. Scheme dependence of Veff is considered. Since no explicit divergences occur when one uses operator regularization, the renormalization group functions (β and γ) associated with the dependence of λ and [Formula: see text] on the radiatively induced scale parameter μ must be determined by considering the finite effective potential. It is shown that one must in fact compute Veff to a higher power in the perturbative expansion than if β and γ were to be computed using Green's functions. The usual results to lowest order are recovered in the [Formula: see text] model. Finally, a nonperturbative β function is determined by requiring that the mass generated by radiative effects be independent of μ2; it is found that both [Formula: see text] and [Formula: see text] are asymptoticly free with this β function. In the appendix we explicitly compute a two-loop integral encountered in the evaluation of Veff.

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Renormalization group determination of the five-loop effective potential for massless scalar field theory

Canadian Journal of Physics ◽

10.1139/p07-202 ◽

2008 ◽

Vol 86 (4) ◽

pp. 623-627 ◽

Cited By ~ 1

Author(s):

F A Chishtie ◽

D G.C. McKeon ◽

T G Steele

Keyword(s):

Field Theory ◽

Renormalization Group ◽

Effective Potential ◽

Massless Scalar ◽

Massless Scalar Field ◽

Minimal Subtraction ◽

Group Methods ◽

Ms Scheme ◽

Renormalization Group Methods

The five-loop effective potential and the associated summation of subleading logarithms for O(4) globally symmetric massless λϕ4 field theory in the Coleman–Weinberg renormalization scheme d4V / dϕ4|ϕ=μ = λ (where μ is the renormalization scale) is calculated via renormalization-group methods. An important aspect of this analysis is conversion of the known five-loop renormalization-group functions in the minimal-subtraction (MS) scheme to the Coleman–Weinberg scheme.PACS Nos.: 11.30.Qc, 11.10.Hi, 11.15.Tk, 12.15.Lk

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The renormalization group functions and the effective potential

Canadian Journal of Physics ◽

10.1139/p94-037 ◽

1994 ◽

Vol 72 (5-6) ◽

pp. 250-257 ◽

Cited By ~ 8

Author(s):

D. G. C. McKeon ◽

A. Kotikov

Keyword(s):

Renormalization Group ◽

Effective Potential ◽

Scale Parameter ◽

Perturbative Expansion ◽

Square Root ◽

Scalar Theory ◽

Higher Power ◽

Text Model ◽

Group Functions ◽

Β Function

It is first demonstrated how the effective potential Veff in a self-interacting scalar theory can be computed using operator regularization. We examine [Formula: see text] and [Formula: see text] theories, recovering the usual results in the former case and showing how Veff is a power series in the square root of the coupling λ in the latter. Scheme dependence of Veff is considered. Since no explicit divergences occur when one uses operator regularization, the renormalization group functions (β and γ) associated with the dependence of λ and [Formula: see text] on the radiatively induced scale parameter μ must be determined by considering the finite effective potential. It is shown that one must in fact compute Vefff to a higher power in the perturbative expansion than if β and γ were to be computed using Green's functions. The usual results to lowest order are recovered in the [Formula: see text] model. Finally, a nonperturbative β function is determined by requiring that the mass generated by radiative effects be independent of μ2; it is found that both [Formula: see text] and [Formula: see text] are asymptoticly free with this β function. In the appendix we explicitly compute a two-loop integral encountered in the evaluation of Veff.

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Zeta-function regularization of quantum field theory

Canadian Journal of Physics ◽

10.1139/p90-093 ◽

1990 ◽

Vol 68 (7-8) ◽

pp. 620-629 ◽

Cited By ~ 8

Author(s):

A. Y. Shiekh

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Renormalization Group ◽

Zeta Function ◽

Analytic Continuation ◽

Quantum Field ◽

Four Dimensions ◽

Group Functions

Analytic continuation leads to the finite renormalization of a quantum field theory. This is illustrated in a determination of the two loop renormalization group functions for [Formula: see text] in four dimensions.

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SUMMING RADIATIVE CORRECTIONS TO THE EFFECTIVE POTENTIAL

International Journal of Modern Physics A ◽

10.1142/s0217751x10051098 ◽

2010 ◽

Vol 25 (31) ◽

pp. 5711-5729 ◽

Cited By ~ 6

Author(s):

F. A. CHISHTIE ◽

T. HANIF ◽

JUNJI JIA ◽

D. G. C. McKEON ◽

T. N. SHERRY

Keyword(s):

Renormalization Group ◽

Effective Potential ◽

Renormalization Group Equation ◽

Group Equation ◽

Renormalization Condition ◽

Group Function ◽

Extremum Condition ◽

Loop Expansion ◽

Group Functions ◽

Renormalization Group Function

When one uses the Coleman–Weinberg renormalization condition, the effective potential V in the massless [Formula: see text] theory with O(N) symmetry is completely determined by the renormalization group functions. It has been shown how the (p + 1) order renormalization group function determine the sum of all the N p LL order contribution to V to all orders in the loop expansion. We discuss here how, in addition to fixing the N p LL contribution to V, the (p + 1) order renormalization group functions can also be used to determine portions of the N p+n LL contributions to V. When these contributions are summed to all orders, the singularity structure of V is altered. An alternate rearrangement of the contributions to V in powers of ln ϕ, when the extremum condition V′(ϕ = v) = 0 is combined with the renormalization group equation, show that either v = 0 or V is independent of ϕ. This conclusion is supported by showing the LL , …, N 4 LL contributions to V become progressively less dependent on ϕ.

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