Axion Models

Prerequisites: Strong CP Puzzle. Full path (41 pages)

Best explanations

Intuitive An Introduction To Axions
Concrete The Strong CP Problem by Pierre Sikivie

Intuitive
Concrete
Abstract
Why is it interesting?
History
Criticism

Intuitive

A great laymen discussion of the axion mechanism is An Introduction To Axions by Susan J Fowler.

One possibility is provided by the fact that the state with $\theta=0$ is the bottom of the zone and thus has the lowest energy. This means if $\theta$ becomes a dynamical variable, the vacuum can relax to the $\theta=0$ state (just like electrons can drop to the bottom of the conduction band by emitting phonons). This is the basis of the axion mechanism [190-192]. The axion is a hypothetical pseudo-scalar particle, which couples to $G \tilde G$ . The equations of motion for the axion field automatically remove the effective $\theta$ term, which is now a combination of $\theta_{QCD}$ and the axion expectation value. Experimental limits on the axion coupling are very severe, but an "invisible axion" might still exist.

page 138 in The QCD Vacuum, Hadrons and Superdense Matter by E. V. Shuryak

Technically this new global U(1) symmetry is not quite an exact symmetry. Like the strong CP symmetry itself, it is a pseudo-symmetry, broken only by non-perturbative or instanton (tunneling) effects. This is exactly why it works as desired. The trick is to make the Higgs field energy depend on the θ value in such a way that, for any initial value of θ, the Higgs field will choose a vacuum value such that the resulting physical parameter θeffective is zero. The Higgs vacuum expectation values aquire phases such that the phase of the quark mass matrix cancels against the initial θ.

http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-8964.pdf

Concrete

QCD allows a term in the Lagrangian that violates P and CP, $\mathcal L_\theta = \bar\theta\,\frac{g_s^2}{32\pi^2}\,G^a_{\mu\nu}\tilde G^{a\,\mu\nu}, \qquad \tilde G^{a\,\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}G^a_{\rho\sigma} ,$ where $\bar\theta$ is the physical combination of the bare $\theta$ angle and the phase of the quark mass determinant. The bound on the neutron electric dipole moment requires $|\bar\theta| \lesssim 10^{-10}$ , while nothing in the Standard Model explains such a small value (see the strong CP puzzle).

The Peccei-Quinn mechanism. Add a new global $U(1)_{PQ}$ symmetry that is spontaneously broken at a high scale and explicitly broken only by the QCD anomaly. Its Goldstone boson is the axion $a$ . Below the breaking scale the QCD anomaly appears as a coupling of the axion to gluons, and the relevant Lagrangian is $\mathcal L = \frac12\partial_\mu a\,\partial^\mu a + \left(\bar\theta + \frac{a}{f_a}\right)\frac{g_s^2}{32\pi^2}\,G^a_{\mu\nu}\tilde G^{a\,\mu\nu} + \dots$ with the axion decay constant $f_a$ . A constant shift $a \to a - \bar\theta f_a$ removes $\bar\theta$ , so the physical question becomes which value of $\langle a\rangle$ the vacuum selects. Nonperturbative QCD effects generate a potential $V(a)$ for the axion that depends on $\bar\theta + a/f_a$ and is minimized where this combination vanishes. The CP-violating angle is thus set to zero dynamically.

Mass and potential. At leading order in chiral perturbation theory with two light quarks, writing $\theta_{\text{eff}} = \bar\theta + a/f_a$ , $V(a) = -m_\pi^2 f_\pi^2\sqrt{1 - \frac{4 m_u m_d}{(m_u + m_d)^2}\,\sin^2\frac{\theta_{\text{eff}}}{2}} ,$ with $f_\pi \approx 92\,\text{MeV}$ . Expanding around the minimum gives the axion mass $m_a^2 = \frac{m_u m_d}{(m_u + m_d)^2}\,\frac{m_\pi^2 f_\pi^2}{f_a^2} ,$ numerically $m_a \approx 5.7\,\mu\text{eV} \times \left(10^{12}\,\text{GeV}/f_a\right)$ . The axion is light and very weakly coupled because all its interactions are suppressed by $1/f_a$ .

Couplings to matter and models. The coupling to photons is $\mathcal L_{a\gamma\gamma} = \frac{g_{a\gamma\gamma}}{4}\,a\,F_{\mu\nu}\tilde F^{\mu\nu} = -g_{a\gamma\gamma}\,a\,\mathbf E\cdot\mathbf B, \qquad g_{a\gamma\gamma} = \frac{\alpha}{2\pi f_a}\left(\frac{E}{N} - 1.92\right) ,$ where $E/N$ is the ratio of the electromagnetic and color anomaly coefficients of $U(1)_{PQ}$ and the number $1.92$ comes from axion-pion mixing. The two benchmark UV completions are the KSVZ model, where the PQ charge sits on new heavy quarks ( $E/N = 0$ in the simplest version), and the DFSZ model, where it sits on two Higgs doublets and the Standard Model quarks ( $E/N = 8/3$ ). Haloscope and helioscope experiments search for axions through the $a\gamma\gamma$ coupling by converting them into photons in strong magnetic fields.

Dark matter. In the early universe the axion field starts at some initial value of $\theta_{\text{eff}}$ . While the Hubble rate is larger than the (temperature-dependent) axion mass the field is frozen. Once $m_a(T)$ exceeds roughly the Hubble rate, the field starts oscillating around the minimum, and these coherent oscillations behave like cold, pressureless matter. This misalignment mechanism makes the axion a dark matter candidate.

A really great and short introduction is The strong CP problem by Pierre Sikivie
Axions and ALPs: a very short introduction by David J. E. Marsh

Abstract

Topology of the $\theta$ term. For an $SU(3)$ gauge field on a closed oriented Euclidean four-manifold (or on $\mathbb R^4$ with finite action), the integral $\nu = \frac{g_s^2}{32\pi^2}\int d^4x\; G^a_{\mu\nu}\tilde G^{a\,\mu\nu}$ equals, up to a sign fixed by orientation conventions, the second Chern number of the gauge bundle and is an integer. The $\theta$ term enters the Euclidean path integral as the phase $e^{i\theta\nu}$ , so $\theta$ is an angle, $\theta \sim \theta + 2\pi$ , and the theory has a family of superselection sectors labeled by $\theta$ . The vacuum energy density $E(\theta)$ is a $2\pi$ -periodic function.

Vafa-Witten argument. In a vector-like gauge theory such as QCD, at $\theta = 0$ the Euclidean measure, including the fermion determinant for positive quark masses, is real and positive. Then $\left|Z(\theta)\right| = \left|\int \mathcal D A\; e^{-S_E}\,\det(\gamma^\mu D_\mu + M)\; e^{i\theta\nu}\right| \le Z(0) ,$ and with $E(\theta) = -\lim_{V\to\infty}\frac1V\ln Z(\theta)$ this gives $E(\theta) \ge E(0)$ . When $\theta$ is promoted to the dynamical field $\theta_{\text{eff}} = \bar\theta + a/f_a$ , the minimum of the axion potential therefore sits at $\theta_{\text{eff}} = 0$ . Expanding near the minimum, the axion mass is fixed by the topological susceptibility of QCD, $m_a^2 f_a^2 = \chi_{\text{top}} = \int d^4x\,\langle q(x)\,q(0)\rangle, \qquad q = \frac{g_s^2}{32\pi^2}G\tilde G ,$ evaluated in Euclidean signature without the axion.

Pseudo-Goldstone structure. The axion is the pseudo-Nambu-Goldstone boson of $U(1)_{PQ}$ . At low energies it is an angular field $a \sim a + 2\pi v_{PQ}$ , where $v_{PQ}$ is the PQ breaking scale. If the PQ charges of the fermions give a color anomaly coefficient $N_{DW}$ (an integer), the potential has period $2\pi f_a$ with $f_a = v_{PQ}/N_{DW}$ . The anomaly breaks $U(1)_{PQ}$ to the discrete subgroup $\mathbb Z_{N_{DW}}$ acting on $a$ , and the vacuum manifold of the potential consists of $N_{DW}$ degenerate minima. This determines the topological defects of axion models: strings from $\pi_1(U(1)) = \mathbb Z$ when PQ breaks, and $N_{DW}$ domain walls attached to each string once the potential turns on. For $N_{DW} > 1$ the resulting network is stable, which is a cosmological problem if PQ breaking happens after inflation.

Shift symmetry and effective theory. Apart from the anomaly coupling, all axion interactions respect the shift symmetry $a \to a + c$ and enter through derivatives, $\frac{\partial_\mu a}{f_a}\bar\psi\gamma^\mu\gamma_5\psi$ , or through anomaly terms with other gauge fields such as $a F\tilde F$ . Particles with these couplings but with mass and decay constant treated as independent parameters are called axion-like particles; they arise generically from compactifications in string theory, where they are the zero modes of higher-form gauge fields.

The QCD axion, precisely by Giovanni Grilli di Cortona, Edward Hardy, Javier Pardo Vega, Giovanni Villadoro

Why is it interesting?

Axions are proposed as a solution to the strong CP problem. In some sense, it's the simplest solution, because the QCD vacuum energy has a minimum at $\bar{\theta}=0$ and the axion allows the vacuum to relax to this ground state. Without the axion the QCD vacuum is frozen and $\bar{\theta}$ is therefore fixed at some not-necessarily minimum value.

History

For a great discussion of the history of axion models, see Axions: Past, Present, and Future by Mark Srednicki

By the time the papers were written Weinberg and Wilczek both used the name “axion” for this particle. I rather liked the alternate name “higglet” which was floating around for a while.

Helen Quinn

Criticism

It may strike the reader (and has struck the author) that enormous theoretical superstructures are here being erected upon a very narrow foundation. The whole superstructure of axions could be made obsolete if a good alternative approach to the problem of strong CP invariance were found. Even if this does happen, I am confident that techniques for dealing with approximate Nambu- Goldstone bosons and their phenomenological (including cosmological) implications will be of enduring interest – so I won't be completely wasting your time.THE U(1) PROBLEM: INSTANTONS, AXIONS, AND FAMILONS by Frank Wilczek

An obvious question about the axion hypothesis is how natural it really is. Why introduce a global PQ “symmetry” if it is not actually a symmetry? What is the sense in constraining a theory so that the classical Lagrangian possesses a certain symmetry if the symmetry is actually anomalous? It could be argued that the best evidence that PQ “symmetries” are natural comes from string theory, which produces them without any contrivance.

Axions In String Theory by Peter Svrcek, Edward Witten

"My favorite solution is that the up quark mass becomes very small by evolving near to a chirally symmetric point. We know that if one of the masses is strictly zero, $\theta$ becomes invisible. The mechanism that I have in mind is that at a very high energy scale, where the weak interactions really communicate a lot with QCD, so that the $\theta$ angle at that scale would be influenced a lot by the weak interactions, one might put one of the quark masses, preferably the up quark mass, to zero. This would correspond to the $U(1)$ chiral symmetry which we know is broken by instantons. So if you scale down from that very high energy scale to low energy scales, the fact that the other quark masses (say the d quark) is unequal to zero will shift the up quark mass to the value it has in the real world. Now to me this appears to be a possible scenario, but it depends very much on numerical analysis whether this works out or not. Usually people who do more detailed calculations tend to disagree with the statement that this is a viable scenario for the $\theta$ angle problem. I am not so totally convinced yet, and I believe that, because of the uncertainties in QCD, this is a possible explanation. It could be for instance, that even if $\theta$ is large at high energies, the renormalization group would rotate it to small values just because the quark masses are so small. The renormalization group could get huge effects from instantons which align according to $\theta$ . I have some hesitation to accept the bold, daring assumptions that e.g. Peccei, Quinn made. See: ’t Hooft

By taking this standard road, one needs to implement the axion in form of a hypothetical degree of freedom from beyond the SM. This requires an introduction of a singlet scalar field with a very large VEV plus either a hypothetical heavy quark [8] or an additional Higgs doublet [9] (for a review see, e.g., [10]).

Domestic Axion Gia Dvali

It is possible that an additional gravitational $\theta$ term:

$\theta_G R \tilde R,$ where $R$ is the Riemann tensor, exists. If this is the case, the axion solution does not work.

there is an old believe that quantum gravity effects can generate an additional breaking of the axion shift symmetry (2) and therefore ruin the axion solution to the strong CP problem (see, e.g., [13]). […] The existence of a nonvanishing topological vacuum susceptibility in pure gravity is currently an open question. […] [I]n the absence of an axion (or a massless fermion), there exist two physically observable theta parameters, one from QCD (1) and one from gravity (9). Consequently, after the axion is introduced, it can only cancel a single combination of the two ϑ-terms, whereas the other combination remains physically observable. Hence, the strong CP problem is not solved.

Domestic Axion Gia Dvali

It is well known that a phase transition with PQ symmetry breaking can form axionic cosmic strings. These strings later become boundaries of domain walls [26] and decay producing axions.

Domestic Axion Gia Dvali

The notion of Peccei–Quinn (PQ) symmetry may seem contrived. Why should there be a U(1) symmetry which is broken at the quantum level but which is exact at the classical level? However, the reason for PQ symmetry may be deeper than we know at present. String theory contains many examples of symmetries which are exact classically but which are broken by quantum anomalies, including PQ symmetry [17–19]. Within field theory, there are examples of theories with automatic PQ symmetry, i.e. where PQ symmetry is a consequence of just the particle content of the theory without adjustment of parameters to special values. The strong CP problem by Pierre Sikivie

There, a global U(1) symmetry (the PQ symmetry) which is almost exact but broken by the axial anomaly of QCD plays a crucial role. After spontaneous breaking, the effective θ-angle of QCD is cancelled by the vacuum expectation value (VEV) of the associated pseudo Nambu-Goldstone boson, the axion a. The origin of such a convenient global symmetry is, however, quite puzzling from the theoretical point of view in many aspects. By definition, the PQ symmetry is not an exact symmetry. Besides, the postulation of global symmetries is not comfortable in the sense of general relativity. It is also argued that all global symmetries are broken by quantum gravity effects [5–10]. If we could regard the PQ symmetry as a U(1) gauge symmetry, there would be no suspicion about the exactness and the consistency with quantum gravity. The PQ symmetry is, however, broken by the QCD anomaly, and hence, it cannot be a consistent gauge symmetry as it is. https://arxiv.org/abs/1703.01112