The Beta-Function Method for Resummations in Field Theory

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


We study the functional integral:
$$\int{g(d\varphi )}\exp \{\int_{\Lambda }{(\lambda :\varphi _{x}^{4}:+\mu :\varphi _{x}^{2}+\alpha :(\partial {{\varphi }_{x}}){{:}^{2}}+\nu )d\xi }\}$$
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:
$$\sum\limits_{\kappa =0}^{N}{{{C}_{\kappa }}}({{\gamma }^{-\kappa }}p){{\gamma }^{-2\kappa }}$$
where N < + ∞ is a cut-off parameter and C κ (p) is holomorphic for|Jp j | < κ, κ > 0, and:
$$\begin{array}{*{20}{c}} {\sum\limits_{\kappa = 0}^\infty {{C_\kappa }} ({\gamma ^{ - \kappa }}p){\gamma ^{ - 2\kappa }} = \frac{1}{{1 + {p^2}}}} \\ {\begin{array}{*{20}{c}} {|{C_\kappa }(p)|\underline < \frac{{{B_\alpha }}}{{1 + |p{|^\alpha }}},}&{\forall \alpha \underline < A,\kappa ,|\mathfrak{J}{p_j}| < \kappa ,A > 8} \end{array}} \end{array}$$


Effective Potential Beta Function Formal Power Series Label Graph Feynman Graph 
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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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