A more effective formulation of the ‘old’ CC problem

A flaw with the previous discussion is that it remains a bit too old school. In particular, it talks about the classical action (and quantum corrections to it) as if Mother Nature gives us a fundamental classical action that we then quantize to make predictions.

But in the modern, more physical, picture of renormalization, there is not really a unique ‘classical’ theory that gets quantized at all. Instead, what one usually calls the classical action is really a fully quantum quantity—the Wilson action—that is appropriate for describing physics below a given energy scale p. In principle, it could be obtained from a more microscopic theory of physics at higher energies by integrating out all of the states having energies larger than p. In this construction, the action S

for the microscopic theory is what we normally call the ‘classical’ action, from which the Wilson action is obtained by integrating out heavy degrees of freedom.^[1]

For example, suppose h^a denotes a collection of quantum fields describing states with energies E > л, and ?^г denotes a collection of quantum fields describing the states with energies E < л. Then the generating functional for correlation functions involving only the light states can be represented in terms of a path integral of the form

where

defines the Wilson action S_W for physics below the scale л. The second line of (4.26) shows that it appears in the expression for e^lW(and so also for observables involving only the ?^г states) in precisely the way a classical action normally would. Furthermore, it can be shown [9] that in many circumstances of interest (i.e. when there is a clear hierarchy of scales between the states described by h^a and those described by A), S_W can be written as an integral over a local Lagrangian density, S_W = f d⁴x L_W, built from polynomials of the fields and their derivatives.

Although the coupling constants appearing in L_W depend explicitly on л this л dependence is guaranteed to cancel with the explicit л dependence associated with the subsequent integration over the light fields A. Notice that this is reminiscent of what happens with the ultraviolet cutoff Л in renormalization theory: the explicit Л dependence arising when integrating out fields cancels the Л dependence that the bare couplings are assumed to have when expressed in terms of renormalized couplings.

Furthermore, the process of ‘integrating out’ heavy states like h^a is clearly recursive, and the Wilson action S_W for another scale Л < Л can be obtained from the Wilson action S_W defined for the scale л by integrating out that subset of A states whose energies lie in the range л < E < л. The new action S_W has an equally good claim to be the ‘classical’ action for observables involving only energies E < л. From this point of view, the original microscopic action S may just be the Wilson action defined at the cutoff scale Л in terms of an even more microscopic action that applies at still higher scales.

When computing only low-energy observables, there is therefore no fundamental reason to call ‘the’ microscopic action S the classical action instead of the Wilson actions S_W or S_W. All have equally good claims, and differ only in the range of scales over which they can be used to compute physical quantities.

Returning now to the ‘old’ cosmological constant problem, although it is possible to absorb the finite m-dependent contributions into a renormalization of the classical action, once this has been done once, it cannot be done again for the effective cosmological constant in the Wilson action at other scales. For which classical action should we do it?

The real answer is that we shouldn’t do it for any of them. What renormalization is telling us is that there is no special classical theory, and so if a physical parameter should happen to be small, its smallness should be understandable in any of the possible effective Wilson actions in which we care to ask the question. Indeed, this is the way hierarchies of scale usually work. For instance, if we wish to understand why atoms are larger than nuclei, we can ask the question in terms of the couplings of the Wilson action appropriate to any scale. If we use the Standard Model, defined at hundreds of GeV, then the size of an atom is set by the Bohr radius, 1/a₀ ~ am_e, and the size of nuclei is set by the QCD scale, 1/r_N ~ Л_дсо. Atoms are larger than nuclei because the fine structure constant is small, a ~ 10~², and the electron is much lighter than the QCD scale, т_е/Л_дси ~ 10~³.

Now suppose we ask the same question in the effective theory below the confinement scale of QCD, where the quarks and gluons of the Standard Model are replaced by the protons and neutrons (or nucleons) that are built out of them. Although the Bohr radius is still set by arm_e in this new theory, the size of a nucleus is now set by the nucleon mass rh_N, where ‘hats’ denote the corresponding renormalized parameters within this new Wilson action. The quantities a, m_e, and rh_N can be computed in terms of the parameters a, m_e, and Л_дсо of the Standard Model, and when this is done, they still satisfy a ^ 1 and rh_e/m_N ^ 1.

In general, there are two parts to understanding why any particular physical parameter (like the cosmological constant or a scalar mass or some other coupling) might be small. One must ask

• 1. Why is the parameter small in the underlying microscopic theory, like S?
• 2. Why does it remain small as one integrates out the higher-energy modes to obtain the Wilson action for the effective theory appropriate to the energies where the parameter is measured?

When both of these questions have an answer, then the small parameter is said to be technically natural. Our understanding of why atoms are large compared with nuclei is technically natural in this sense. Unfortunately, both of these questions do not have an answer for the cosmological constant within the Standard Model, and so our understanding of the small size of this parameter is not technically natural.^[2]

In the effective theory well below the milli-eV scale, which is implicitly used in cosmology, the effective cosmological constant can indeed be renormalized to be of the order of the observed dark energy density: Vo_jle — p_vac — (10~² eV)⁴. But this then also tells us how big the cosmological constant, Vo_jhe, must be in the Wilson action for the effective theory above the electron mass. Because the electron is present in this high-energy theory, but not in the lower-energy one at sub-eV energies,^[3] Vo_jle and Vo_jhe must be related by a formula like (4.24):

The only way Vo _le can be small enough is to have Vo _he and the electron quantum correction cancel one another to an accuracy of better than 30 decimal places! And the required cancellation only gets worse as one moves to the Wilson action defined at energies above the masses of even heavier particles.

We know of no other hierarchies of scale in Nature that work like this, and that is the more precise reason why predictions like (4.24) or (4.28) are really a problem.

Solving this problem is clearly going to be hard; it involves modifying how even the electron contribution to the vacuum energy gravitates, and moreover this must be done at low energies right down to sub-eV scales. But the electron is probably the particle that we think we understand the best, and this modification must be done in such a way as not to ruin any of the many successful predictions that have been made of electron properties at these energies. This seems a tall order (though it may yet be possible, as argued below).

On the other hand, the upside to the need to modify physics at very low energies is that any successful proposal is likely to be testable in many ways using a variety of low-energy experiments. If there were only at least one solution to the problem, it would be very predictive. In practice, it has not yet been possible to profit from this observation in the absence of any convincing candidate solutions (more about which below).

• [1] The description here follows that of the reviews [9].
• [2] There is another, slightly more specific, criterion which is sometimes known as ‘technical naturalness’, which is phrased in terms of symmetries. This other criterion is here called’t Hooft naturalness,and is described in more detail in a later section. So, in the terminology of this chapter, ’t Hooftnaturalness is sufficient for technical naturalness, but needn’t be equivalent to it. Supersymmetrictheories can have small parameters that are ‘supernatural’, that is are technically natural though notprotected by symmetries (and so not ’t Hooft natural).
• [3] More properly, because the electron is stable, it can be present in the low-energy theory byrestricting to states having a definite lepton number. But its antiparticle, the positron, is integratedout in this effective theory, precluding there being large quantum corrections to the vacuum energyfrom electron-positron loops.