Why is it correct to estimate divergences by the cutoff in QFT?

Let's say we have a linear divergence in a quantum field theory. The way to deal with this infinite quantum correction is to go through the whole process of renormalization.

However, quite often, people estimate the order of magnitude of this divergence by just substituting the upper limit of the integral with the energy scale when our theory breaks down.

My question is, why is this a legit thing to do? I do not see any connection at all with this estimation and with the correct way of dealing with these infinities via renormalization (where there is no cutoff, just a regulator that is taken to infinity in the end anyway).

• I think you're confused about some things there. A cutoff is a regulator, and when considering the behavior of the theory when the regulator (i.e. the cutoff) goes to infinity one is effectively estimating the divergences in the QFT. – David Vercauteren Apr 21 '15 at 9:44
• @David: Well, first there is the cutoff that one uses as a regulator, which indeed goes to infinity. But then there is also the cutoff that sets the scale when one expects the theory to become invalid. – Jonathan Lindgren Apr 29 '15 at 12:49
• @David: Well, first there is the cutoff that one uses as a regulator, which indeed goes to infinity. But then there is also the "cutoff" that sets the scale when one expects the theory to become invalid. But sometimes people take this second cutoff, and substitutes it in the divergences, and claims that this estimates the quantum correction. This is what I do not understand. – Jonathan Lindgren Apr 29 '15 at 12:54
• Mathematically these two types of cutoffs are identical. The only difference is the physical meaning. When people are interested in divergences, they want to know how the theory behaves mathematically, so it doesn't matter at all whether you interpret the cutoff as being a maximal scale of the theory or as just a regulator for the integrations. – David Vercauteren May 5 '15 at 8:48

There are two different ways of dealing with infinities:

1. There is renormalization which has nothing to do with substituting some value for the cutoff. Rather, parameters of the Lagrangian are expressed in terms of measurable quantities which effectively hides the divergences.

2. Another way of trying to get information out of these infinities is to treat a quantum field theory with divergences as an effective theory which is valid up to some scale. One typically assumes that the theory at the high scale has couplings of order of the cutoff scale which get corrected by the renormalisation group flow from the high scales down to the scales at which the effective theory is valid. Integrating quantum corrections up to the cutoff scale basically amounts to removing all the corrections and getting an expression for the parameters at the high scale theory in terms of the low scale parameters.

Let me give you two examples of that thinking: The first example is the statement of the infamous 120 order of magnitude difference between the estimated value of the vacuum energy and the measured cosmological constant. To get the estimated value of the vacuum energy one assumes that QFT is valid up to the Planck scale and ignores physics beyond the cutoff. The contributions from divergent loop integrals in bubble diagrams are then integrated up to the Planck scale instead of being renormalized away. The value of vacuum energy which effectively would act as a cosmological constant is off by many orders of magnitude, which leads to the cosmological constant problem.

Another example is the naturalness problem of the standard model. Here, one again assumes that the Standard model is valid up to some scale where new physics kicks in. We can then do loop calculations in QFT but instead of renormalizing, we look at the SM parameters -- in this case the Higgs mass -- as a function of the cutoff and value of the parameter at that cutoff. If one now requires that the parameter reproduces the known Higgs mass at low energies and the cutoff is such that the higgs mass is of about order one at the cutoff. This yields a value around the TeV scale and is the reason why particle physicists likes to have susy around the TeV scale.

There are many variations of the latter argument which are called hierarchy problem (if we take the cutoff all the way to the Planck scale than we would find that the Higgs should be much heavier), or the fine-tuning problem (if the cutoff is at the Planck mass, then we need to fine-tune our parameters such that we get the known Higgs mass)

The bottom line is that we know that in many condensed matter systems this logic is applicable: We have a theory at a high cutoff (typically of the order of angstrom where the atomic theory is valid). There, parameters are of order one and get corrected by thermal and quantum fluctuations as we go to larger distances. Vice-versa, if we start with an effective theory of macroscopic phenomena and integrate corrections up to some cutoff scale, we find that parameters become natural at the angstrom scale (we can effectively predict the atomic scale!) This is the justification to do this.