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What is sector decomposition and how can it be used to 'disentangle' UV and IR divergences?

I have read about it in the paper SecDec: A general program for sector decomposition,
but I have no idea if, with a suitable change of variable $x=1/q$ on each coordinate, we can change the IR divergences to UV divergences or change the UV divergences to IR ones and then apply sector method to disentangle the divergences.

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2 Answers

UV divergence is due to self-action. It exists even in statics. No self-action is introduced, no UV divergence appears in calculation. Renormalizations discard the self-action contributions perturbatively rather than exactly, at the beginning.

IR divergence is due to too distant initial approximation, distant from the exact solution, of course. It is like using a Taylor expansion $f(x) = f(0) + f'(0)\cdot x + ...$ for calculations where $x$ is infinite (distant from zero).

So these divergences are not interchangeable.

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First paragraph is correct, the second one is not. IR divergences are due to singularities near the massless shell $p^2 \to 0$ and they are quite similar to UV divergencies which appear as $p^2 \to \infty$. In particular, it seems that they can be swapped by $p^2 \leftrightarrow {1 \over p^2}$, precisely as Jose claims. I am not familiar with that method though. –  Marek Aug 1 '11 at 5:44
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But anyway, thanks for an honest attempt at addressing the question. I like that you are finally only referring to the content of the question and not to problems of renormalization as a whole. –  Marek Aug 1 '11 at 5:45
    
@Marek: I think that the variables the OP referred to are Feynman parameters, not momenta. And although you can make the change $x\to1/x$ it does not change the type of divergence. –  Simon Aug 1 '11 at 6:11
    
@Simon: oh, I see. That is a different question altogether. –  Marek Aug 1 '11 at 6:18
    
@Marek: Marek, try to understand the simple problem of coupling presented in my talk docs.google.com/… You are sufficiently qualified to understand the physical motivation and technical implementation of the problem. Maybe then you will see me and my answers differently. –  Vladimir Kalitvianski Aug 1 '11 at 9:36
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Disclaimer: I've never done any real work with sector decomposition - I've only read (some of) the papers and played with some simple examples.


There are many different sector decomposition strategies, which are mainly designed to work on Feynman parametrized integrals. In general, sector decomposition is a algorithmic (recursive or not) way of decomposing Feynman parametrized integrals into subsectors where the divergences can be more easily isolated. Traditionally it was used in formal investigations and proofs relating to BPHZ, nowadays it is often used for automated separation of divergences ($\epsilon$-expansion) and numerical integration for comparison with analytic results and for use where analytic results are not known.

Sector decomposition does not mix up UV and IR divergences. However, some strategies and changes of variables can make it difficult to easily tell them apart. Also, if dimensional regularization is the only regularization used, then both UV and IR divergences turn up as poles in the $\epsilon$-expansion. If care is not taken to track where the poles come from, then you can not separate the UV and IR contributions. There are sector decomposition schemes, e.g. arXiv:0908.2897, that take care with the IR divergences. Also see older work such as PhysRevD.10.3991.

To clarify this (hopefully), let's give a quick summary of parametrized Feynman integrals:


There are two main parameterized versions of Feynman integrals (both of which can be derived from the position and momentum space versions). For a quick discussion see Scalar Feynman Diagrams and Symanzik Polynomials. For a longer discussion see the paper linked to in the question and references within.

The Symanzik polynomials $\mathcal{U}$ and $\mathcal{V}$ are key aspects of both Feynman parameterization (Chisholm representation)

Feynman parameters ......... (1)

and Schwinger parametrization (aka propertime or $\alpha$ parameters) (Nambu representation).

Schwinger parameters ......................... (2)

Feynman diagrams with tensor, spinor or other structures eventually lead to similar integrals, but with extra polynomial factors in the numerator.

In the Schwinger parametrization (2), UV divergences are zeros of the 1st Symanzik/Kirchhoff polynomial $\mathcal{U}(s)$ that occur when various subsets of $s_i$ are zero. IR divergences come from zeros of exponential (which includes the 2nd Symanzik polynomial $\mathcal{V}(s,p_{\text{ext}})$) as a subset of $s_i$ go to infinity. UV divergences only depend on the topology of the diagram , but since $\mathcal{V}$ has dependence on the external momenta, IR divergences can also depend on the kinematics.

Similar statements hold for the Feynman parametrization (1), but once the $\delta$ function is integrated out it can be confusing as to whether the singularities in the integrand are UV or IR.


To quote Hepp and Speer sectors within modern strategies of sector decomposition

To analyze convergence of the integral (2) large values of $s_i$ (in particular, to reveal infrared (IR) divergences) one has to take into account the exponent as well. A possible way is to separate the integration over every $s_i$ into (0, 1) and (1,∞) and then to deal with each of these $2n$ regions separately. This can be enough for a general analysis but cannot be good from the practical point of view because the number of the resulting sectors will be too large. A more reasonable approach is to turn (2) into the integral (1) with a compact integration domain, where both UV and IR divergences are somehow mixed up and manifest themselves as divergences at small values of parameters of integration.

This said, Feynman parameterization and sector decomposition don't actually mix UV and IR divergences, it can just make them difficult to naively identify. Also, both turn up as poles in the $\epsilon$-expansion, so can be hard to separate after the integral has been performed.

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