This statement applies to the potential, but in general one must also grapple with low multiplicity derivative interactions that are set by the scale $\mathrm{\Lambda }$ without any additional suppression by $\epsilon$. In this case the true cutoff $\mathrm{\Lambda }$ is accessible at low multiplicity and there is no mirage. For example, unsuppressed interactions of this kind appear in many UV completions of the axion after integrating out an accompanying radial Higgs mode or tower of extradimensional excitations. Still, it is still striking that the axion potential exhibits a mirage cutoff.

Alternatively, we can enforce a boundary on field space by multiplying the kinetic energy term by a step function $\theta (\overline{\varphi }-\varphi )$. The physical equivalence between an infinite wall in the potential energy versus a step function in the kinetic energy is analogous to what happens in Einstein versus Jordan frame in gravity. For a detailed review of the EFT of phonons in a physical bar, see Appendix pp1. In Appendix pp2 we also present a simple model of balls, springs, and strings whose mean field approximation exhibits a boundary in field space.

Notably, boundaries in coordinate space have long appeared in the context of gravity as geometric cutoffs or “Dirichlet walls.” Since these boundaries are not dynamical branes, they need not be stabilized, but at the same time their consistency is not guaranteed. Nevertheless, under certain conditions, these endpoint objects have been found to be stable [5, 6, 7] and consistent with the initial value problem [8].

Here we define our potential to be a series expansion in $({\varphi }^2/{\mathrm{\Lambda }}^2{)}^n$ with coefficients that scale as $1/n!$, as opposed to a series expansion in $(\varphi /\mathrm{\Lambda }{)}^n$ with coefficients that scale as $1/n!$. We do this for two reasons. First, our phenomenological applications all involve field space boundaries for charged fields like the Higgs boson or the SM force carriers, which only appear in even powers in their self-interactions. The second reason has to do with a minor subtlety regarding the scale of unitarity violation. A general potential defined by a series expansion in $(\varphi /\mathrm{\Lambda }{)}^n$ with coefficients $1/n!$ will cover the familiar case of a periodic scalar, as well as the exponential runaway of a dilaton field. However, as shown in [2], these models exhibit scattering which at any multiplicity will never see a breakdown of the EFT at the scale $\mathrm{\Lambda }$ but rather at the slightly higher cutoff $\mathrm{\Lambda }\sqrt{\log (1/\epsilon )}$, which can be realized in explicit UV models [2]. Of course, these factors of $\log (1/\epsilon )$ corrections are far less important than the powers of $\epsilon$ that generate the phenomenon of the mirage cutoff. Nevertheless, for simplicity we restrict here to a potential which is a series expansion in $({\varphi }^2/{\mathrm{\Lambda }}^2{)}^n$ with coefficients $1/n!$, which as shown in Appendix pp3 has a physical cutoff defined by unitarity violation in scattering that is precisely the scale $\mathrm{\Lambda }$, with no additional factors of $\log (1/\epsilon )$. Note that in general, changing the power of $\varphi$ in the series expansion has the same effect as changing the scaling of the series coefficient. So for example, a series expansion in $({\varphi }^k/{\mathrm{\Lambda }}^k{)}^n$ with coefficients $1/n!$ is the same as a series expansion in $(\varphi /\mathrm{\Lambda }{)}^n$ with coefficients $(1/n!{)}^{1/k}$.

It may be concerning that this potential—like that of the axion—is an entire function. In particular, there is evidence that the generating functional of connected correlators $W[J]$ cannot be entire [11, 12], which is related to the asymptotic nature of the perturbative series [13, 14, 15, 16]. Furthermore, in certain explicit examples the operator product expansion can also be asymptotic [17]. Nevertheless, we note that these degenerations at high multiplicity all stem from the growth of diagrammatic topologies in correlators, while the potential defines individual interaction vertices. As shown in Appendix pp3, any contribution to a correlators involving multiple insertions of the potential interactions will be subleading since $\epsilon \ll 1$. Consequently, the growth in diagrams is tamed and any subtleties concerning the asymptotic nature of perturbation theory are not relevant.

As shown in Sec. 4, radiative corrections serve only to renormalize the coefficients $c_n$ to numerical values of the same order. Since this manifestly preserves the structure of the potential in Eq. (5), it also maintains the flatness of the potential for $|\varphi |\lesssim \overline{\varphi }$. Nevertheless, these corrections could in principle still flip the signs of $c_n$ in such a way that destabilizes the potential, toggling the feature at $|\varphi |\sim \overline{\varphi }$ from a wall into a cliff. This is reminiscent of the sensitivity of vacuum alignment to quantum corrections, for example as occurs in radiative electroweak symmetry breaking. Here also, the binary question of whether the potential develops a wall or a cliff will be decided by the UV dynamics that regulate the divergences, so in this sense these two possibilities have even odds. Notably, if the underlying theory is supersymmetric, then the potential energy will be automatically bounded from below.

One should bear in mind that the EFT transitions from the weak field to strong field regime long before $\varphi$ reaches $\overline{\varphi }$. To see why, recall that the weak field regime by definition applies when perturbation theory in the field value is accurate. This implies that the $n$-point interaction in $V(\varphi )$ must be more important than the ($n+1$)-point interaction. From $V(\varphi )$ it is clear that this transition occurs at the critical field value $|\varphi |\sim \mathrm{\Lambda }\ll \overline{\varphi }$.

The potential terms generate derivative interactions in the same way as in ${\varphi }^4$ theory, where wave function renormalization appears at two loops.

What is the natural size for these derivatively coupled interactions? Consider, for example, the $(\partial \varphi {)}^4$ operator, whose coefficient is famously constrained to be nonnegative by analyticity, unitarity, and causality [22]. In particular, this coefficient is equal to an energy weighted integral over the total cross-section for $\varphi \varphi \to \chi$, where $\chi$ denotes the heavy states in the UV completion. To preserve the phenomenon of the mirage cutoff, the coefficient of $(\partial \varphi {)}^4$ must be suppressed by $\epsilon \ll 1$, which implies that $\chi$ must couple very weakly to a pair of $\varphi$ particles. This is very different than what happens in traditional UV completions, like the linear sigma model completion of a periodic scalar. Conversely, an EFT with a mirage cutoff can only arise in a UV completion in which the heavy state $\chi$ only couples to a high multiplicity of $\varphi$ particles.

If the mass of $\varphi$ is calculable as a function of the parameters of the UV theory, then the dependence on $\tilde{\mathrm{\Lambda }}$ will emerge from the physical mass scale of the heavy degrees of freedom. This happens, for example, in phenomenological models of soft supersymmetry breaking. If the mass of $\varphi$ is incalculable, however, then its quadratic divergences must canceled by a counterterm whose value is uniquely fixed by the physical mass, which can only be extracted from experimental observations. In this case, the mass cannot be predicted in terms of UV parameters, and technically there is no hierarchy problem.

Is there a version of electroweak symmetry breaking that directly utilizes a boundary in Higgs field space? The simplest version of this idea is to set $\lambda =0$, so the tachyonic mass for the Higgs field drives its VEV to the boundary of field space. For $\mathrm{\Lambda }\ll \overline{H}$, which is required for radiative stability of the boundary, the VEV is pushed to the edge, $⟨H⟩\sim \overline{H}$, as expected, irrespective of the value of tachyonic mass $m^2$. Unfortunately, by expanding in fluctuations, $H=⟨H⟩+h$, we see that the Higgs particle exhibits nonlinear interactions that enter in powers of $h\overline{H}/{\mathrm{\Lambda }}^2$, which are enhanced for $\mathrm{\Lambda }\ll \overline{H}$. Since the Higgs VEV is pinned inside the cusp between the tachyonic mass and the boundary, the Higgs particle is strongly coupled and thus not viable.

We have imposed a boundary on the electromagnetic field so as to preserve gauge invariance. In principle, we could instead consider an infinite wall boundary on the gauge field itself, corresponding to a Proca potential which has been sequestered to high multiplicity. If we regulate the boundary wall to have finite thickness, then tiny but nonzero Proca potential terms will be induced in the trivial vacuum. While such a setup might be viable for an Abelian gauge field like hypercharge, this is likely pathological for the non-Abelian case, where Proca mass terms invariably lower the cutoff.

Strictly speaking, the number of photons produced in a given scattering process is ill-defined on account of infrared divergences. However, since the field strength associated with a soft photon is vanishingly small, it essentially decouples from all higher dimension operators. This accords with the calculation in Appendix pp5, which shows that EFT breaks down when the energy per particle exceeds the cutoff.