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In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. One traditional technique is to modify the integral by adding a ghost. Therefore, the above integral can be modified to

$$\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}-\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-\Lambda^{2})^{2}}\sim i \log\frac{m^2}{\Lambda^2}$$

This defines a curve in the complex plane $\Lambda^{2}\rightarrow i \log\frac{m^2}{\Lambda^2}$

So we don't associate a finite complex number to the integral, but we associate a curve to the integral. My question is: Do mathematicians study divergent integrals in this way? By generalizing the notion of integration, such that it is possible to be valued in the space of curves in the complex plane?

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Trying to assign a value to one single divergent integral is usually meaningless. What does make sense however is to try to assign a value to a very large collection of divergent integrals in a consistent way. Here, "consistent" should be interpreted along the lines of "in such a way that all exact identities between these integrals that should formally hold do actually hold".

There are various ways of doing this, but as far as I am aware, they all boil down to a variant of the following procedure.

  • Find a (linear) space $T$ that indexes your collection of "divergent integrals". This is typically some space of Feynman diagrams, maybe with additional decorations.

  • Find a space $\mathcal{M}$ of linear maps $\Pi \colon T \to A$ for some space $A$, which should be thought of as all "plausible" ways of assigning a value to your integrals. The definition of $\mathcal{M}$ should enforce the "consistency" mentioned above. (For example, $T$ usually has an algebra structure in which case the same should be true of $A$ and $\Pi$ should be an algebra homomorphism, but one may want to impose additional constraints.) It is natural to take $A = \mathbf{R}$ or $A = \mathbf{C}$, but other choices can be natural depending on your renormalisation procedure, provided that you have a natural projection $\pi \colon A \to \mathbf{C}$. For Feynman diagrams with "legs", it would be natural to take for $A$ a space of distributions.

  • Find a group $\mathfrak{R}$ acting on $T$ in such a way that $\Pi \mapsto \Pi \circ M$ preserves $\mathcal{M}$. The Hopf algebra showing up in Connes-Kreimer and mentioned by @gmvh would be the universal enveloping algebra of $\mathcal{R}$. (Or maybe its dual.)

  • Fix some regularisation procedure that gives you a family of valuations $\Pi_\varepsilon \in \mathcal{M}$ which can honestly be claimed to approximate the integrals you want to approximate. In the case of Connes-Kreimer, they take dimensional regularisation, which yields a Laurent series in $\varepsilon$ for every integral. They would then view this as a single valuation with values in the space of Laurent series $A = \mathbf{C}\{\{z\}\}$, and as approximating the projection $\pi$ (evaluation at $z = 0$) by $\pi_\varepsilon$ (evaluation at $z = \varepsilon$).

  • Show that you can find a sequence of elements $M_\varepsilon \in \mathfrak{R}$ such that $\Pi_\varepsilon \circ M_\varepsilon$ (or $\pi_\varepsilon \circ \Pi_\varepsilon \circ M_\varepsilon$ if you take the viewpoint that it is the projection that is being approximated) converges to a limit: that's the consistent valuation you're after.

This procedure is pretty general and not restricted to dimensional regularisation at all: you can find an implementation with $A$ a bona fide space of distributions (the product is the tensor product) and quite arbitrary regularisations in this article of mine. As presented here, this procedure does not give you a canonical valuation since the limit is only unique up to an element of $\mathfrak{R}$. You may then want to impose further constraints in an ad hoc manner to enforce uniqueness. For example, you can enforce that irreducible vacuum diagrams evaluate to $0$ (BPHZ prescription).

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In order to render divergent Feynman integrals meaningful, some prescription must be used.

The one generally used in physics is to make the integral finite by imposing a regulator (such as the Pauli-Villars regulator you quote, or dimensional regularization, or even a brute cut-off) and then to absorb the regulator dependence into a renormalization prescription.

More mathematical approaches to the problem (which at some level boil down to the same thing as the physics approach, but in a more mathematically structured way) can be found in the work of people such as Dirk Kreimer, where Feynman diagrams are related to elements of a Hopf algebra. In such an approach, the regularized Feynman rules correspond to an algebra homomorphism from that Hopf algebra into the algebra of the objects that "try to be" distributions on external leg data (test functions) for the diagrams, but actually map only into $\mathbb{C}\{\{z\}\}$ (cf. the PhD thesis of Susama Agarwala) -- that is the closest to your idea that mathematicians study to my knowledge.

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These integrals do not make much sense in mathematics. So, what you wrote is meaningless. The only reasonable approach is to integrate on a finite interval with a cut-off so to have a finite integral. Then, if the theory is renormalizable, you will realize that all these contributions from the cut-off entail a redefinition of the constants of the theory and you can complete your computation with such "renormalized" constants.

By the way, it is possible that some future theory will provide finite results, until then we can content ourselves with a not-too-much rigorous procedure without pretending a mathematical justification. This will apply also to the dimensional regularization technique. I do not even know if any attempt has been ever pursued to justify it mathematically.

Last but non least, the current formulations of our quantum field theories largely use Feynman path integrals that have not got any serious mathematical justification for their existence. So, we are in the unpleasant situation to do "renormalization" on something that is not proven to exist. We have just the comfort that we are able to get experimental results with very high precision with such completely unjustified procedure.

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