Abstract
General description of ultraviolet divergences in the \(\phi ^4_4\) model. Loop and one-particle irreducible (1PI) diagrams. The superficial degree of divergence. Renormalization of the one-loop contribution to the four-point Green’s function (the sunset diagram). The BPHZ subtraction scheme. Lorentz invariant renormalization of the two-point Green’s function. The renormalization constants \(Z_1, Z_3, \delta m^2\) and the multiplicative renormalization.
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Notes
- 1.
In the rather general discussion below, we neglect numerical factors which are present in the perturbative contributions, because they are not important in the qualitative analysis of the UV divergences.
- 2.
Let us recall that in calculus, the integrals of the type \(\int _{-\infty }^{+\infty } \) are defined as the limit of \(\int ^{M_2}_{M_1}\) when \(M_1 \rightarrow - \infty , \;\; M_2 \rightarrow + \infty .\)
- 3.
If this assumption is abandoned the graphs can be drawn in the plane, see Exercise 8.1.
- 4.
The integrals may still be divergent for specific values of the external momenta, because the denominators of some propagators can be equal to zero. In order to avoid such divergences, we may replace \(i0_+\) in the denominators by \(i \epsilon \), where \(\epsilon >0.\) The limit \(\epsilon \rightarrow 0_+\) is taken after we perform the integrations over loop momenta. \(\tilde{G}^{(n)}(k_1, k_2, \ldots , k_n)\) is not a smooth function of the external momenta—rather, it is a generalized function of them. Singularities of these functions usually have certain physical meaning. We shall not discuss them because their presence does not jeopardize the existence of the perturbative contributions.
- 5.
In some rather special cases the integral can be finite even if \(\omega \ge 0\), because the integral over the solid angle \(\Omega \) can vanish. We shall not consider such exceptions.
- 6.
We denote the graph and the formula corresponding to it by the same letter. The presence of P–V regularization is marked by adding the argument M.
- 7.
One may remove more terms than necessary. Such an operation is called an oversubtraction.
- 8.
This series is likely not convergent. Typically, one expects that perturbative expansions in quantum field theory yield a so called asymptotic series which form a special class of divergent series. In most applications of the perturbative expansions, the series is either cut to a finite sum of graphs (then the problem of convergence disappears), or it is restricted to an infinite subclass of graphs which are distinguished by their particularly simple analytical contributions (and then sometimes one can compute the sum).
- 9.
The factor 1/2 is canceled by the combinatorial factor 2, which appears because the vertex can be connected to two lines in two ways.
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Arodź, H., Hadasz, L. (2017). Renormalization. In: Lectures on Classical and Quantum Theory of Fields. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-55619-2_8
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DOI: https://doi.org/10.1007/978-3-319-55619-2_8
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