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The Beta-Function Method for Resummations in Field Theory

  • Giovanni Gallavotti
Part of the NATO ASI Series book series (NSSB, volume 234)

Abstract

We study the functional integral:
$$\int{g(d\varphi )}\exp \{\int_{\Lambda }{(\lambda :\varphi _{x}^{4}:+\mu :\varphi _{x}^{2}+\alpha :(\partial {{\varphi }_{x}}){{:}^{2}}+\nu )d\xi }\}$$
(1.1)
where Λ is a d-dimensional torus, λ, μ, α, ν are constants and g is a gaussian random field over Λ with a covariance obtained by periodizing the function on R 4 whose Fourier transform is:
$$\sum\limits_{\kappa =0}^{N}{{{C}_{\kappa }}}({{\gamma }^{-\kappa }}p){{\gamma }^{-2\kappa }}$$
(1.2)
where N < + ∞ is a cut-off parameter and C κ (p) is holomorphic for|Jp j | < κ, κ > 0, and:
$$\begin{array}{*{20}{c}} {\sum\limits_{\kappa = 0}^\infty {{C_\kappa }} ({\gamma ^{ - \kappa }}p){\gamma ^{ - 2\kappa }} = \frac{1}{{1 + {p^2}}}} \\ {\begin{array}{*{20}{c}} {|{C_\kappa }(p)|\underline < \frac{{{B_\alpha }}}{{1 + |p{|^\alpha }}},}&{\forall \alpha \underline < A,\kappa ,|\mathfrak{J}{p_j}| < \kappa ,A > 8} \end{array}} \end{array}$$
(1.3)

Keywords

Effective Potential Beta Function Formal Power Series Label Graph Feynman Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Giovanni Gallavotti
    • 1
  1. 1.Centro Interdisciplinare Linceo B. SegreAccademis dei LinceiRomaItaly

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