UV and IR Effects in Axion Quality Control

Low-energy scalars typically emerge in the 4D effective theory from such fields in one of two ways:

• •

  $T$-type axions: $b(x)$ are specific cases of Kaluza-Klein (KK) modes arising when dimensionally reducing the extra-dimensional components $B_{mn}(x,y)=b(x)\,\omega_{mn}(y)$, where $x^{\mu}$ denote the observed 4 dimensions, $y^{m}$ are extra-dimensional coordinates and $\omega_{mn}(y)$ is a harmonic 2-form field within the extra dimensions.

• •

  $S$-type axions: ${a}(x)$ arise directly as the 4-dimensional components $B_{\mu\nu}(x,y)=b_{\mu\nu}(x)\,\omega(y)$, which in four dimensions are known to be dual to scalar fields with shift symmetries Savit:1979ny through relations of the form $\partial^{\mu}{a}\propto\epsilon^{\mu\nu\lambda\rho}\partial_{\nu}B_{\lambda\rho}$ (much more about which below). Here $\omega(y)$ is a harmonic 0-form field – typically a $y$-independent constant that can depend on extra-dimensional moduli.

This UV provenance is of course relevant to the axion quality problem, which is in essence an issue of UV sensitivity. One of our goals with this paper is to explore the ways that it helps, for both $T$- and $S$-type axions. Some of our conclusions are similar to earlier discussions of this issue Kallosh:1995hi ; DualStrongCP , in particular that the problem gets rephrased in dual form (for $S$-type axions) in terms of the existence of multiple 3-form gauge potentials.

Since these issues have recently been revisited anew Sakhelashvili:2021eid ; Dvali:2022fdv we clarify what properties these fields must have to actually cause a quality problem and use this to argue why gravitational examples specifically (and the great abundance of such potentials in string vacua more generally) need not pose a problem in themselves. The dual formulation also suggests how the presence of multiple axions (as is common in string theory) can help alleviate the quality problem. The upshot is that the UV can, but need not, cause a quality problem. Whether or not it does cannot be decided purely at low energies because it depends on what happens in the UV.⁴⁴4The same is is also true of other naturalness problems; they arise because of strong dependence on physical masses for states that actually appear in the UV theory and not a dependence on cutoffs, as is sometimes mistakenly asserted (for a summary of these issues see e.g. Burgess:2013ara ).

But our discussion has implications that apply more broadly than just to the quality problem for the QCD axion. Along the way we identify more generally how dimensional ‘naturalness’ arguments for the scalar potential give very different estimates depending on whether they are made directly for the scalar or are first done for its dual and then mapped to the scalar using duality. In particular terms involving $n$ powers of the canonically normalized scalar arise additionally suppressed by powers of $(m/M)^{n}$ where $M$ is the UV scale and $m$ is the axion mass (an observation also made in the past for inflationary models NaturalnessForms ).

We find a number of other ways that axion properties suggested by string-motivated extra-dimensional physics can be informative. For instance we describe a simple model for which $T$-type axions have physical axion-matter couplings $g_{{a\hbox{\kern-0.70004pt}f\hbox{\kern-0.70004pt}f}}$ that are dramatically smaller than the naive value $1/f$ read off from the axion kinetic term. In the example given (motivated by the models of YogaDE ) $g_{{a\hbox{\kern-0.70004pt}f\hbox{\kern-0.70004pt}f}}$ is order $1/M_{p}$ despite $f$ being an ordinary particle-physics scale. Decoupling these scales from one another could have practical implications for axion phenomenology.

We show why the same hierarchy does not arise in these models for $S$-type axions and we clarify why not. Physical couplings of $S$-type axions really are of order $1/f$ and we identify which interactions in the UV completion are responsible for the breakdown of the $E/f$ expansion at energies $E\mathrel{\raise 1.29167pt\hbox{$>$\kern-8.50006pt\lower 4.30554pt\hbox{$\sim$}}}f$. $S$-type axions illustrate how the scalar and dual representations can provide instances of weak/strong coupling duality, for which both the scalar and the dual cannot be within the weakly coupled regime. In the extra-dimensional example studied it is the Kalb-Ramond formulation that is weakly coupled. This could also have phenomenological implications to the extent that an axion that is dual to a weakly coupled system is unlikely to be well-described by the semiclassical methods that are universally used when exploring its physical implications.

Some of these observations imply that the use of the scalar (rather than Kalb-Ramond) variable can be misleading in some circumstances. This can seem surprising at first sight because the duality between axions and Kalb-Ramond fields is in essence a field redefinition and so scalar and dual formulations should be completely equivalent; it shouldn’t matter that string theory hands you Kalb-Ramond fields if scalar axions are equivalent and are much simpler to work with. Why should one care that a more complicated framework exists if it only obscures implications drawn using more transparent methods? We argue here that phenomena like weak/strong coupling duality are special cases of Weinberg’s Third Law of Progress in Theoretical Physics Weinberg:1981qq : You can use any degrees of freedom you like to describe a physical system, but if you use the wrong ones you’ll be sorry.

These duality arguments touch on a related rich vein of physics with broader significance: the importance of keeping non-propagating entities like auxiliary and/or topological fields when formulating Wilsonian effective theories. These are fields that can be integrated out without changing the types of particles that propagate, and so it is tempting to think one should do so once and for all and simply ignore them thereafter. However such fields bring to the low-energy effective theory information about how its UV completion responds, e.g. to environments with nontrivial topology. They arise in concrete situations (such as in EFTs for 3-dimensional Quantum Hall systems, where the presence of emergent non-propagating gauge fields is essential for capturing the fractional quantization of Hall plateaux and the unusual charge and statistics of some excitations QHE ; EFTBook ).

Evidence is building that a similar role is played more widely by 3-form gauge potentials in four spacetime dimensions, $C:=\frac{1}{6}\,C_{\mu\nu\lambda}\,{\rm d}x^{\mu}{\rm d}x^{\nu}{\rm d}x^{\lambda}$ subject to the gauge freedom $C\to C+{\rm d}\Lambda$ where $\Lambda_{\mu\nu}(x)=-\Lambda_{\nu\mu}(x)$ is an arbitrary 2-form field. These are known to bring to the low-energy 4D effective theory topological information coming from integrated out extra dimensions Bousso:2000xa ; Burgess:2015lda , and more generally provide the origin for the auxiliary fields that appear in the 4D supergravities that are the low-energy limits of string vacua Bielleman:2015ina ; Herraez:2018vae . They appear in the QCD quality story because they can give masses to Kalb-Ramond fields Quevedo:1996uu through a Higgs mechanism that is dual to more mundane methods of axion mass generation. Because the field strength $H={\rm d}C$ often appears in the action with a definite sign (often as a square), its presence can alter the implications of naturalness arguments for the scalar potential Burgess:2021juk . Indeed such terms provide the 4D understanding of why 6D SLED models SLED can in some circumstances suppress the 4D vacuum energy, but also why they struggle to do so enough to solve the cosmological constant problem Burgess:2015gba ; Burgess:2015lda ; Burgess:2015kda ; Niedermann:2015via . Their interplay with accidental scaling symmetries lies behind a recent attempt to find a dynamical relaxation mechanism for vacuum energies in four dimensions YogaDE .

In what follows we build our case for the above story using concrete examples. We first, in §2, briefly review the duality construction – in particular its extension to massive axions Quevedo:1996uu , which provides a way to think about scalar masses arising through a Higgs mechanism. §3 then briefly reviews and clarifies its use to dualize the axion solution to the strong-CP problem DualStrongCP , highlighting in particular how the quality problem gets rephrased in the dual language and how the apparent UV sensitivity of terms in the EFT differs between the axion formulation and its dual. Finally §4 provides a concrete extra-dimensional example – inspired by a UV completion of YogaDE – that illustrates both how axion/Kalb-Ramond duality can map weak to strong couplings, and how enormous hierarchies can arise with $g_{{a\hbox{\kern-0.70004pt}f\hbox{\kern-0.70004pt}f}},g_{a\gamma\gamma}\sim 1/M_{p}$ even with $f$ as low as eV scales.

Consider the following path integral

where $S_{1}=\int{\rm d}^{4}x\;{\cal L}_{1}$ with ${\cal L}_{1}$ chosen (at least to start) to be

with $G={\rm d}B$ the exterior derivative of a 2-form field $B_{\mu\nu}$ and ${\cal Z}$ and $J_{\rho}$ possibly depending on other fields (collectively denoted $\psi$). $B=\frac{1}{2}\,B_{\mu\nu}\,{\rm d}x^{\mu}\wedge{\rm d}x^{\nu}$ is only defined up to the gauge redundancy $B\to B+{\rm d}\lambda$ for an arbitrary 1-form $\lambda=\lambda_{\mu}\,{\rm d}x^{\mu}$.

The duality starts by trading the integration over $B_{\mu\nu}$ for an integral over $G_{\mu\nu\lambda}$ subject to a constraint that imposes the Bianchi identity ${\rm d}G=0$. These are equivalent because the Bianchi identity is sufficient to guarantee the local existence of a field $B_{\mu\nu}$ with $G={\rm d}B$. The constraint is imposed by integrating over a scalar Lagrange-multiplier field ${a}$, and so writing

where $S_{0}=\int{\rm d}^{4}x\;{\cal L}_{0}$ with

Integrating out ${a}$ imposes the Bianchi identity ${\rm d}G=0$ and allows the integral over $G$ to be replaced with the integral over $B$, leading back to (2).

The dual version is obtained from (4) by instead integrating out $G_{\mu\nu\lambda}$ so that ${a}$ is the remaining field. The result inherits a shift symmetry ${a}\to{a}\;+$ constant because ${\cal L}_{0}$ transforms into a total derivative. The $G$ integration is gaussian, whose saddle point is $G_{\mu\nu\lambda}={\cal G}_{\mu\nu\lambda}$ where

and so the integration gives the new lagrangian density

If ${\cal Z}=1$ then ${a}$ is a canonically normalized massless scalar derivatively coupled to the same local current $J_{\mu}$ as in the original formulation. Because (2) and (6) are both obtained from (4) they must describe equivalent physics. Although the implied field redefinition from $B_{\mu\nu}$ to ${a}$ is in principle nonlocal the physics on both sides is nonetheless local because this is true of the relation between the field strengths given in (5).

Although the above makes the shift symmetry (and so also masslessness) of ${a}$ seem automatic, we next summarize how duality extends to massive scalars, following Quevedo:1996uu . A scalar potential is achieved in the dual framing through a Higgs mechanism in which the field $B_{\mu\nu}$ ‘eats’ (or is eaten by) a non-propagating gauge potential⁶⁶6Known string vacua can also contain a large number of these 3-form gauge potentials. $C_{\mu\nu\lambda}$. Because $C_{\mu\nu\lambda}$ does not propagate this meal does not change the number of propagating degrees of freedom.

To this end consider the following gaussian path integral

where $S_{1}=\int{\rm d}^{4}x\;{\cal L}_{1}$ and

Here $C_{\mu\nu\lambda}$ is a 3-form gauge potential with field strength $H={\rm d}C$ and $B_{\mu\nu}$ is a 2-form gauge potential with $G={\rm d}B$ while $m$ is a parameter with dimension mass.

This lagrangian has the gauge symmetry $C\to C+{\rm d}\Lambda$ and $B\to B-m\,\Lambda$ for an arbitrary 2-form $\Lambda$. So when $m\neq 0$ we can choose a gauge $B=0$. The field equation for $C$ that follows from this action then is

This describes a single spin state propagating with mass $m$ once all the gauge symmetries are used, as can be seen by counting the massless states from which it is built. (In 4D $B_{\mu\nu}$ is shown above to be equivalent to a massless scalar and $C_{\mu\nu\lambda}$ contains no propagating degrees of freedom at all because one can always write $H_{\mu\nu\lambda\rho}=h\,\epsilon_{\mu\nu\lambda\rho}$ with field equation $\partial_{\mu}H^{\mu\nu\lambda\rho}=0$ in the massless limit, which implies $h$ is a constant and so does not propagate.)

The dual should therefore be a massive scalar and this can be verified by trading the integral over $B$ for an integral over $G$ and introducing (as before) a lagrange multiplier ${a}$ to impose the Bianchi identity⁷⁷7One can equivalently omit the $mC_{\mu\nu\lambda}$ terms everywhere and instead impose the modified Bianchi identity ${\rm d}G=mH$. ${\rm d}G=0$, leading to the lagrangian density

Integrating out ${a}$ returns us to the above formulation, but instead performing the integration over $G$ leads to the saddle point

and so to the lagrangian

Next we perform the integral over $C_{\mu\nu\lambda}$, and this is equivalent to simply performing the gaussian integral over $H_{\mu\nu\lambda\rho}$ because the integrability condition for writing $H={\rm d}C$ is ${\rm d}H=0$ which is always true (in 4D). The saddle point for the $H$ integral occurs for $H_{\mu\nu\lambda\rho}={\cal H}_{\mu\nu\lambda\rho}$ where

and so leads to the scalar lagrangian

This is the expected massive scalar.

To this end we extend the above reasoning to the main event: QCD and the $\theta$-term. The idea is to dualize the coupling of the axion to QCD to see how the strong-CP problem gets formulated, along the general lines of DualStrongCP . We then ask how UV physics might complicate the story in the dual theory. Consider then adding a gauge potential $A_{\mu}$ (with field strength $F_{\mu\nu}$) to represent the QCD gauge sector⁸⁸8We do not write quarks explicitly but flag the few places where their implicit presence affects what is written. and this time consider the path integral

where $S_{0}=\int{\rm d}^{4}x\;{\cal L}_{0}$ and

We suppress both gauge-group indices and traces over them to avoid notational clutter. $F_{\mu\nu}$ is the field strength for the gauge potential $A_{\mu}$ but $G_{\mu\nu\lambda}$ is an arbitrary 3-form until the integral over ${a}$ is performed.

Integrating out ${a}$ imposes the Bianchi identity ${\rm d}G=\Omega$ where $\Omega$ is a gauge-invariant quantity built from the gauge field that on grounds of consistency must satisfy ${\rm d}\Omega=0$, for which we take

The mass scale $f$ is here required on dimensional grounds. Doing this allows the $G$ integral to be traded for one over $B$ as before and gives the lagrangian

where $G={\rm d}B+S$ where ${\rm d}\Omega=0$ implies there locally exists an $S_{\mu\nu\lambda}$ – the Chern-Simons 3-form – that satisfies $\Omega={\rm d}S$.

The dual formulation instead integrates out $G$ and leaves ${a}$ as the dual variable. Integrating out $G$ leads to the lagrangian density

This shows that the standard axion-gauge coupling is the dual of the 2-form/QCD coupling given in ${\cal L}_{1}$ and that $f$ can be interpreted as its decay constant.

We now have the tools required to explore UV sensitivity and the axion quality problem. We start by restating the original formulaton of the quality problem and then how it is rephrased in 2-form language for both $T$-type (this section) and $S$-type (next section) axions.

The axion quality problem asks two related questions QualityProblem :

1. 1.

   Do corrections to the QCD axion potential change its minimum in a way that preserves a sufficiently small effective vacuum angle: $\bar{\theta}_{\rm eff}\lesssim 10^{-10}$?

2. 2.

   Do corrections to the QCD axion potential change the usual expression for the axion mass (that assumes it is dominantly generated by the ‘IR-dominated’ QCD instanton with size $\rho\sim\Lambda_{{\scriptscriptstyle Q\hbox{\kern-0.50003pt}C\hbox{\kern-0.50003pt}D}}^{-1}$)?

The first of these essentially asks if the QCD axion remains a good solution to the strong CP problem when perturbed by new physics, whereas the second asks the same of our understanding of axion mass. The axion mass question can apply more generally to ALPs as well, whereas the first one is specific to the QCD axion.

Any UV completion must decide what happens at energies above the axion decay constant $f$ above which the low-energy expansion in powers of $E/f$ breaks down. We consider in turn the original formulation and the $T$- and $S$-type axions that arise within an extra-dimensional context.

For $S$-type axions the representation directly obtained from UV physics is the field $b_{\mu\nu}$ dual to the scalar axion. And as alluded to earlier – c.f. §2.2.1 – issues of UV sensitivity can look very different in dual formulations to scalar theories, with for example the existence of a dual implying that the effective couplings for terms like ${a}^{n}\in V_{\scriptscriptstyle U\hbox{\kern-0.50003pt}V}({a})$ come suppressed by powers of the axion mass $(m/M)^{n}$ relative to generic scalar estimates. Such suppressions can be enormous given the small size of $m$ relative to UV scales.

We therefore revisit earlier discussions of how the axion quality problem arises in the dual formulation, partly motivated by recent discussions Sakhelashvili:2021eid ; Dvali:2022fdv that argue that gravity causes new problems. Although we confirm the important role played by multiple 3-form potentials DualStrongCP in the framing of the dual quality problem, we also show that the many 3-forms found in string vacua do not generically pose a problem. Problems are only caused where strongly interacting systems make instanton-like effects important and this is not the case for the many ‘elementary’ 3-forms that descend from extra dimensional vacua. We argue that for similar reasons 4D gravitational Chern-Simons forms also need not cause problems (such as for string vacua where the UV completion of gravity is described by weakly coupled physics).

To the extent that the shape of the axion potential $V({a})$ is dual to interactions like $W(X)$ involving the 4-form field strength $X=\frac{1}{4!}H_{\mu\nu\lambda\rho}\epsilon^{\mu\nu\lambda\rho}$, one might think that the dual version of the axion quality issue should hinge on the detailed form of UV contributions to $W(X)$. This proves not to be right, as we now argue. The central point turns on the Legendre transformation relating $V({a})$ to $W(X)$; in particular on (32) and (34), that state

On the scalar side the strong-CP problem is not solved unless $m{a}={\overline{\theta}}\tilde{\Lambda}^{2}$ at the minimum of $V$, and the quality problem is the statement that corrections to $V$ can perturb the minimum so that this relation fails. Although $X$ always vanishes at a minimum for $V$, eq. (41) suggests that on the dual side the criterion for satisfying the strong-CP problem is that $\partial W/\partial X=0$ is satisfied when $X=0$. So the quality problem seems to hinge on whether or not UV physics can introduce a linear term $\delta W=\eta X$ whose inclusion would modify (41) in a way that obstructs having $m\,{a}={\overline{\theta}}\tilde{\Lambda}_{\scriptscriptstyle Q\hbox{\kern-0.50003pt}C\hbox{\kern-0.50003pt}D}^{2}$ be a solution to $\partial V/\partial{a}=0$.

Suppose, then, that one finds after integrating out the UV physics an EFT below the QCD scale of the form (27), but with a linear term in $X$ whose coefficient is not proportional to the CP violating parameter ${\overline{\theta}}$: