We live in a Universe where the rules of quantum mechanics reign. Even though we usually associate the quantum world with the microscopic world, it is not true that quantum mechanics bears no consequences on physical phenomenon on astrophysical or cosmological scales. Such an example is that of astrophysical neutrinos. Humanity can now directly verify that the Sun runs off nuclear reactions driven by the weak force, because there are several detectors around the world sensitive to neutrinos generated deep within the core of our parent star as the result of its dominant pp cycle. However, due to quantum mechanics — and despite the neutrinos propagating over the macroscopic Earth-Sun distance — there is a non-zero probability that the original electron-type neutrinos engendered within the Sun are found instead as muon-type or tau-type neutrinos here on Earth.

Motivated by the cosmological constant problem and the discovery that the universe is experiencing accelerated expansion, many cosmologists spend their time studying alternate theories of gravitation that oftentimes involve adding to Einstein’s General Relativity scalar fields that couple — as universally as scalar fields can — to ordinary matter. One possible goal of these theories is to do away with the need to introduce a cosmological constant term in Einstein’s equations, because many Quantum Field Theorists find its presence un-`natural’. In particular, the measured value of the (square root of the) cosmological constant , assuming it accounts for all of Dark Energy, is significantly smaller than the energy scales set by the fundamental microscopic interactions among elementary particles. However, because General Relativity is well tested from millimeter to Solar System length scales, there is a need to screen (i.e., make as weak as possible) such a hypothetical `5th force’ in Nature mediated by one of these scalar fields, so that they are not already ruled out by available data.

It turns out there are several such candidate theories on the market today. To screen the force it mediates, near the source of gravity, such a scalar field typically acquires one of the following characteristics: (I) grow large relative to some energy scale , namely, ; or (II) its gradients grow large relative to , namely and/or . (See for instance Section 3.2 of arXiv: 1407.0059 and 2.2.4 of arXiv: 1601.06133.) Because these field theories are designed to drive the observed acceleration of the universe, a cosmological phenomenon, the length scale occurring within them — i.e., the — are often macroscopic ones. This makes contact with the quantum nature of the Universe we live in, because the scalar self-interactions responsible for such a supposed screening effect, involving , and , are often irrelevant in the Quantum Field Theory (QFT) sense. As I will now argue qualitatively (and non-rigorously), this means quantum effects might be so important over macroscopic scales that the very mechanism invented to screen the 5th force cannot be relied on to work as intended.

**Vainshtein & Galileons** For concreteness, let’s examine the Galileons, a class of scalar field theories that *Vainshtein screen* the 5th force it mediates by having the second derivatives be large, specifically . About a 4-dimensional (4D) Minkowskian background, their dynamics is described by the following Lagrangian density:

(1):

In 4D flat spacetime, an `irrelevant’ term in a Lorentz invariant QFT has coefficients that scale as . This indicates the interaction terms in eq. (1) are irrelevant; note that I have displayed them in schematic form because my arguments below do not for the most part depend on their details. Their detailed form — see equations (34) through (38) of arXiv: 0811.2197 — *does* ensure that the associated equation-of-motion (EoM) of the Galileon is second-order in time derivatives, so it does not suffer from Ostrogradsky instability (see arXiv:1506.02210 for a review): non-degenerate higher order Lagrangians would yield Hamiltonians that are linear in the highest-order momentum, and are therefore unbounded from below. (In plain language: the energy of systems suffering from the instability can — pathologically! — approach negative infinity unhindered, usually by imparting infinite positive energy to other systems it interacts with.)

The primary issue with `irrelevant’ terms in a field theory’s dynamics is that, once they become necessary to describe the physics at hand, you’d need to include an increasing number of them the more accurately you want to model your physical system. In fact, the rules of perturbative QFT tell us one may have no choice. Even if you forgot to include certain irrelevant terms in your Lagrangian density, the results of perturbative QFT calculations — because of the quantum nature of our world and the self-interactions of the screened scalar field — would become infinite. And as a result, irrelevant terms involving higher powers of 1/Mass would have to be added as counter-terms to cancel these infinities. For the Galileon, these considerations yield at the very least the following infinite series, which I again present in schematic form to emphasize that this is a series involving the power counting parameter :

(2):

where all the are dimension-less pure numbers. There are other terms such as

(2′): , and

(See arXiv: 1310.0187 for a pure Galileon quantum correction calculation.) Now, the expert reader may complain I have ignored in eq. (2) the “Galilean” symmetry enjoyed by the Galileon — the invariance, up to a total divergence, of the Lagrangian density in eq. (1) under the replacement , where and are constants. However, this symmetry is no longer respected by the matter-Galileon coupling (see for e.g., the in eq. (3) below) nor in curved spacetime (see arXiv: 0901.1314). As such, I do not believe one needs to regard it as fundamental.

While there are non-renormalization theorems guaranteeing that do not themselves receive any quantum corrections, I believe the same theorems do not prohibit the existence of non-zero . It is also my understanding that these non-renormalization theorems were proven without considering interactions with generic forms of matter; but Galileons, being a 5th force that “modifies” the equivalence-principle respecting gravitational interaction, really should couple to *all* matter. Since , this indicates we likely have in eq. (2) a non-convergent or very slowly converging series precisely where we need the screening to be effective. In other words, our Galileon theory cannot provide a sensible example of Vainshtein screening because it cannot be trusted precisely where we need its self-interactions to substantially weaken the Galileon force relative to that of gravity itself. A physical parallel of this Galileon situation, is to employ the Euler-Heisenberg effective dynamics to model quantum electrodynamics at energy scales significantly higher than the electron mass. Or, use the Fermi theory to describe electroweak interactions at energies far beyond the and boson masses. A simple mathematical analogy is to attempt to evaluate at say using its geometric series representation .

One may try to salvage this Galileon theory by imagining that the could turn out to be tiny enough to render the harmless; this is of course a distinct possibility but remember, though, that under renormalization group flow, the infinite number of coupling constants in eq. (2) do not generically remain zero unless some additional physical principle is uncovered/introduced.

**Length scales** For Galileons to be relevant to cosmic acceleration, cosmologists usually set the Galileon energy scale based on Newton’s gravitational constant , as well as the current Hubble parameter (a measure of the Universe’s current rate of expansion):

You might think this indicates the theory ceases to be viable below length scales of a thousand kilometers or so, already a rather discouraging state of affairs on its own. The situation turns out to be far more dire, however. To be more specific, let us follow the literature and consider the lowest order coupling to ordinary matter:

(3):

where is the trace of the energy-momentum-stress-shear tensor of some astrophysical system. (As an aside, observe the Galileon does not couple to the photon at this order since the latter’s stress tensor is traceless; i.e., it violates the equivalence principle.) Modeling the Sun as a point mass ; placing it at the origin of our coordinate system; and ignoring the planets for now — a possibly very poor assumption given how important the nonlinearities will turn out to be — we find that

(4):

Using 3D spherical coordinates , the ODE that arises from solving in equations (1), (3), and (4) then reads:

(5):

The detailed structure of the Galileon interaction (see discussion on p11 of arXiv:0811.2197) is responsible for the term dropping out for a time-independent profile. (Also, to arrive at eq. (5), it is useful to recognize that the `left-hand-side’ of the equations-of-motion for Galileons take the `total-divergence’ form , for an appropriately defined . This allows Gauss’ theorem to be applied, reducing the EoMs to first order ones. On the other hand, this feature is unlikely to continue to hold once the terms in eq. (2) are included.) The in eq. (5) tell us the higher powers of are directly inherited from the nonlinearities of the Galileon theory. Heuristically speaking, if we associate every derivative with one power of , then . We may use this to estimate eq. (5) to go as

(5′):

where is the Vainshtein radius of the Sun:

(6):

As one gets close to the Sun — i.e., as grows — we expect the highest order term to dominate:

Referring to eq. (6), we see that our power counting parameter is not only greater than one, it is *much* greater than unity for the inner Solar System. The Earth is roughly *light minutes* from the Sun, compared to the hundreds of *light years* Vainshtein radius. Hence,

This teaches us that the series in eq. (2) likely does not converge, or does so very slowly, once light years — instead of the original expectation of km. In the same vein, this result that indicates the quantum corrections from pure Galileon self-interactions, i.e., the terms in eq. (2′), are so large that the entire theory very likely ceases to be valid below . For example, the following series is probably very ill defined, or at least not useful at all for making physical predictions, within our Solar System:

Now, to be fair, this power counting parameter does get a tad smaller the higher the powers of we include. That is, let us consider modifying eq. (5′) as follows:

Near the Sun, where the highest power term dominates, we have

(7):

which means as . I’m not sure, though, if this inspires confidence in the utility of the series theory of eq. (2) — for, even if it converges, one still has an infinite number of coefficients to match against Nature, unless some underlying organizing principle is uncovered.

**Vainshtein Screening** On the other hand, let us verify that the Vainshtein screening mechanism *does* work, provided we are willing to suspend our doubts about the validity of theory itself. Taking eq. (7) into account, the Galileon force per unit mass due to the Sun is

Whereas the Newtonian gravity force per unit mass due to the Sun is . Taking the ratio of the Galileon to Newtonian gravity force laws, we verify that the Galileon force is indeed suppressed near the Sun:

**Superluminality** Finally, I should mention: small waves propagating on top of its spherically symmetric solution sourced by a central mass can travel faster than the speed of light. This means the chronology of cause versus effect is ambiguous in these theories, even if one believes no closed timelike curves can be set up.

**Skepticism** I have not examined too closely the other screening mechanisms on the market, involving and . However, I cannot help but express concerns of the same spirit as the Galileon case: why wouldn’t one face a non-convergent or very slowly converging series in either or ?

I end with a question that sums up my skepticism:

Is it really possible to implement a mechanism for weakening a hypothetical `5th force’ in Nature near its material source, that involves only a small number of large self-interactions and that is simultaneously relevant for cosmic acceleration?

**Update 5 November 2017:** To improve the precision of the arguments, I have added the phrase “very slowly converging” to my discussions on the “non-convergent” Galileon theories. Compare, for instance, to the first terms of its Taylor series . For large , one would have to sum up many terms, i.e., would have to be very large, in order to accurately capture the behavior of . I’m not sure, however, if it is possible to prove rigorously that the Galileon field theory (or the Euler-Heisenberg, or Fermi 4-fermion theory) necessarily *has* to break down beyond their relevant energy scales — even if physical evidence strongly suggests they do — but it is reasonable to suspect, at the very least, the resulting series is very slowly converging and therefore useless for making physical predictions.

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