The axion is a hypothetical type of field/fundamental particle originally hypothesized (Weinberg 77, Wilcek 78) as a solution to the strong CP problem in the standard model of particle physics. After the initial model for the axion was quickly ruled out by experiment (see Wilcek 78 for the early history) a variant model was found (Dine-Fischler-Srednicki 81), called the “invisible axion”, which does not violate experimental bounds.

The “invisible axion” turns out to also be a natural candidate for dark matter Preskill-Wise-Wilcek 83, hence potentially also solves one of the problems with the standard model of cosmology. More recently it is argued in Hui-Ostriker-Tremaine-Witten 16 that indeed axionic dark matter (fuzzy dark matter) potentially solves the remaining problems of standard WIMP dark matter models: WIMP dark matter models work exceedingly well on cosmological scales but has serious experimental problems on the scale of galaxies. Due to the extreme lightness of axion particles, their de Broglie wavelength may be of the scale of galaxies and hence their quantum properties may become relevant at this scale to deviate in the right way from the WIMP models.

The Pecchei-Quinn shift symmetry of the axion and the peculiar nature of the axion interaction term which are needed to make the axion model work this way have been argued to naturally arise in string theory, if the axion is identified with the KK-reduction of the higher gauge fields in string theory. This we discuss below.

As a solution to the strong CP problem

The problem here is that the action functional for QCD (Yang-Mills theory) a priori contains an arbitrary theta angle θ\theta

S:1g 2 Xtr(F F )+θ Xtr(F F ) S \;\colon\; \nabla \mapsto \frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla) \;+\; \theta \int_X tr(F_\nabla \wedge F_\nabla)

but that – since the term tr(F F )tr(F_\nabla \wedge F_\nabla) causes parity violation, which is strongly constrained by experiment – there must be some reason why θ\theta vanishes or else is extremely small.

(Here the term tr(F F )tr(F_\nabla \wedge F_\nabla) is the invariant polynomial for the second Chern class, measuring the instanton number of the gauge field \nabla.)

The solution to this problem via axions is to assume that θ\theta is not really a fundamental parameter, but instead is the vacuum expectation value of a dynamical field aa (the axion).

The idea is that a standard kinetic action

S kin(a) Xa*a S_{kin} (a) \propto \int_X a \wedge \ast a

together with the axion interaction term

S int(a,) Xatr(F F ) S_{int} (a,\nabla) \propto \int_X a \, tr(F_\nabla \wedge F_\nabla)

makes aa have vanishing expectation value a\langle a\rangle. This would give a dynamical explanation why under the identification of the theta angle with this expectation value

θa \theta \coloneqq \langle a\rangle

the theta angle vanishes.

The Vafa-Witten mechanism

That this is indeed the case is due to Vafa-Witten 84, around (2). These authors argue via the Wick rotated path integral as follows:

Under Wick rotation, parity-violating terms in the Lagrangian density pick up an imaginary factor ii. Therefore the path integral expression for the Wick rotated vacuum energy is

E vac(a) =log(,aexp(S(,a))DDa) =log(exp(1g 2 Xtr(F F ))Daexp(ia Xtr(F F ))Da). \begin{aligned} E_{vac}(a) &= - log \left( \underset{\nabla,a}{\int} \exp(- S(\nabla,a)) \, D \nabla \, D a \right) \\ &= - log \left( \underset{\nabla}{\int} \exp\left( \frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla) \right) \, D \nabla \; \underset{a}{\int} \exp\left( i a \int_X tr(F_\nabla \wedge F_\nabla) \right) \, D a \right) \end{aligned} \,.

Now due to the factor of ii in front of θ\theta in this expression, the real part of the argument of the logarithm necessarily becomes smaller with aa. Therefore the negative logarithm becomes larger with aa. Accordingly E vac(a)E_{vac}(a) must have a minimum at θ=0\theta = 0 (according to Vafa-Witten 84, p.2).

In string theory

In string theory axion fields

  1. are naturally present as the KK-reduction of the higher gauge fields (e.g. Svrcek-Witten 06);

    reviewed below in:

    As KK-Reduction of higher gauge fields

  2. are generically of positive but tiny mass (ADDKM 09, Acharya-Bobkov-Kumar 10) as they need to be to be a solution to the strong CP problem and be candidates for BEC/superfluid/fuzzy dark matter;

    reviewed below in:

    Stringy axion phenomenology

This makes axion fields in string theory form a curious confluence point relating


higher gauge theory particle physics/cosmology phenomenology higher gauge fields higher characteristic classes non-perturbative QFT/string effects in HET: Green-Schwarz anomaly cancellation in IIA/B: higher WZW term for Green-Scharz D-branes} axion fields in the string spectrum {solve strong CP-problem as with P-Q robustly solve dark matter problem by FDM \array{ \mathbf{\text{higher gauge theory}} && && \mathbf{\text{particle physics/cosmology phenomenology}} \\ \\ \left. \array{ \text{higher gauge fields} \\ \text{higher characteristic classes} \\ \updownarrow \\ \text{non-perturbative QFT/string effects} \\ \text{in HET: Green-Schwarz anomaly cancellation} \\ \text{in IIA/B: higher WZW term for Green-Scharz D-branes} } \right\} &\longrightarrow& \array{ \text{axion fields} \\ \text{in the string spectrum} } &\longrightarrow& \left\{ \array{ \text{solve strong CP-problem as with P-Q robustly} \\ \text{solve dark matter problem by FDM} } \right. }

Axions as KK-reduction of higher gauge fields

In string theory axion fields naturally arise as the KK compactification of the higher gauge fields (B-field, C-field, RR-fields). Here we review how this comes about.

Notice that this means that axions in string theory are as in the output of the original Peccei–Quinn proposal, but do not actually make use of the Peccei–Quinn mechanism.

As amplified in particular in Svrcek-Witten 06:

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. … the string compactifications always generate PQ symmetries, often spontaneously broken at the string scale and producing axions, but sometimes unbroken. (Svrcek-Witten 06, pages 3-4)

Similarly from ADDKM 09:

However, at the effective field theory level it is hard to judge how natural it is to have such a “fake” global PQ symmetry which is explicitly broken exclusively by QCD. Note, that in order not to spoil the solution to the strong CP problem all other sources of explicit PQ symmetry breaking should be at least 10 orders of magnitude down with respect to the potential generated by the QCD anomaly, and one may be especially cautious about the viability of such a proposal, given the common lore that global symmetries always get broken by quantum gravitational effects [12, 13, 14]. This makes it natural to inquire whether axions arise naturally in the most developed quantum theory of gravity—string theory. (ADDKM 09, p. 3)

The mechanism discussed in Svrcek-Witten 06, in all its incarnations in the various perspectives on string theory (heterotic strings, type II strings, 11-dimensional supergravity with M-theory effects included, etc.) share the following properties:

  1. the axion field itself is the double dimensional reduction of one of the higher gauge fields in string theory, the (twisted) Kalb-Ramond field for heterotic string, the RR-field for the type II string, or the supergravity C-field in 11-dimensional supergravity with M-brane effects;

  2. accordingly the Peccei-Quinn periodic shift symmetry of the axion results arises as the U(1)U(1)-gauge symmetry that is the dimensional reduction of the B 2U(1)B^2 U(1) (heterotic B-field) or B 3U(1)B^3 U(1) (11d C-field) higher gauge symmetry, where the U(1)U(1)-periodicity is ultimately due to the higher Dirac charge quantization for these fields;

  3. the axion interaction term of the form aFF\propto a \langle F \wedge F\rangle arises as the double dimensional reduction of the self-interaction terms of these higher gauge fields:

We now discuss this in more detail:

In type IIA string theory

Consider string phenomenology in type IIA string theory in the guise of intersecting D-brane models with the standard model of particle physics supported in intersecting D6-branes whose worldvolume fills all of 4d spacetime.

The higher WZW term in the Green-Schwarz action functional for the D6-brane is of the form

X 7A RRexp(F ) \propto \int_{X_7} A^RR \wedge \langle \exp(F_\nabla) \rangle

where A RRA^RR denotes the collective inhomogenous background RR-field and F F_\nabla is the curvature 2-form of the Chan-Paton gauge field on the D-brane. The \langle -\rangle denotes the Killing form invariant polynomial, hence the trace for the special unitary Lie algebra regarded as a matrix Lie algebra. X 7X_7 denotes the worldvolume of the D6-brane

Hence one of the three non-vanishing summands in this expression is

X 7A 3 RRF F . \propto \int_{X_7} A_3^{RR} \wedge \langle F_\nabla \wedge F_\nabla\rangle \,.

Now assuming a Kaluza-Klein ansatz X 7=X 4×Y 3X_7 = X_4 \times Y_3 with

a Y 3A 3 RR a \coloneqq \int_{Y_3} A_3^{RR}

the effective axion field in 4d, then this term becomes the axion interaction term

X 4aF F \propto \int_{X_4} a \langle F_{\nabla} \wedge F_{\nabla}\rangle

(Svrcek-Witten 06, around (6.9))

In heterotic string theory

In heterotic string theory KK-compactified to 4d, the 4d B-field, dualized, serves as the axion field, called the “model independent axion” (Svrcek-Witten 06, starting p. 15).

This works as follows: By the Green-Schwarz anomaly cancellation mechanism, then B-field in heterotic string theory is a twisted 2-form field, whose field strength HH locally has in addition to the exact differential dBd B also a fundamental 3-form contribution, so that

H=dB+C H = d B + C

(locally). Moreover, the differential dHd H is constrained to be the Pontryagin 4-form of the gauge potential \nabla (minus that of the Riemann curvature, but let’s suppress this notationally for the present purpose):

dH=tr(F F ). d H = tr \left(F_\nabla \wedge F_\nabla\right) \,.

Now suppose KK-compactification to 4d has been taken care of, then this constraint may be implemented in the equations of motion by adding it to the action functional, multiplied with a Lagrange multiplier :

S= XHHkinetic actionfor B-field+ Xa(dHtr(F F ))Green-Schwarz constraint. S = \underset{ \text{kinetic action} \atop \text{for B-field} }{ \underbrace{\int_X H \wedge \star H} } + \underset{ \text{Green-Schwarz constraint} }{ \underbrace{ \int_X a \left( d H - tr(F_\nabla \wedge F_\nabla) \right) } } \,.

Now by the usual argument, one says that instead of varying by aa and thus implementing the Green-Schwarz anomaly cancellation constraint, it is equivalent to first vary with respect to the other fields, and then insert the resulting equations in terms of aa into the action functional.

Now since we are dealing with a twisted B-field, with free 3-form component CC, we actually vary with respect to CC. This yields the Euler-Lagrange equation of motion

da=H, d a = \star H \,,

hence the usual relation or electro-magnetic duality, expressing what used to be the Lagrange multiplier now as the magentic dual field to the twisted B-field.

Plugging this algebraic equation of motion back into the above action functional for HH gives

S˜= Xdadakinetic actionfor axion field+ Xatr(F F )axionicinteraction. \tilde S = \underset{\text{kinetic action} \atop \text{for axion field}}{\underbrace{\int_X d a \wedge \star d a}} + \underset{\text{axionic} \atop \text{interaction}}{\underbrace{\int_X a \, tr(F_\nabla \wedge F_\nabla)}} \,.

This now is an action functional for an axion field aa of just the form required above for the solution of the strong CP-problem.

Stringy axion phenomenology

From Acharya-Kane-Kumar 12, p. 3:

Now consider the axions. Due to the shift symmetries mentioned above, there are no perturbative contributions to their potential. Non-perturbative effects though, such as strong gauge dynamics, gauge instantons, gaugino condensation and stringy instantons will generate a potential for the axions. Because any such contribution is exponentially suppressed by couplings and/or extra-dimensional volumes, in our world with perturbative gauge couplings (at high scales), the axion masses will be exponentially small.

Furthermore, since there are large numbers of axions in general, their masses will essentially be uniformly distributed on a logarithmic scale (5). See (6) for a detailed calculation of axion masses in string/M effective theories.

Like the moduli, the axions are also very weakly coupled to matter and therefore do not thermalize in general. Moreover, since their masses are tiny - ranging from m 3/2m_{3/2} to even below the Hubble scale today - many of them, including the QCD axion, start coherent oscillations during the time that the moduli are dominating the energy density, but before BBN.

Observe that ultralight axion fields is precisely what is required in HEP phenomenology for

  1. the solution to the strong CP-problem;

  2. the solution of the dark matter problem via BEC/superfluid/wave/fuzzy dark matter (e.g. Hui-Ostriker-Tremaine-Witten 16).



The axion as such was originally proposed in

  • Steven Weinberg, New light boson, Phys. Rev. Lett. 40:223-6 (1977)

  • Frank Wilcek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40:279-82 (1978)

The experimentally viable variant as the “invisible axion” is due to

The observation that the “invisible axion” is a candidate for dark matter is due to three groups:

  • John Preskill, M. Wise, Frank Wilcek, Cosmology of the invisible axion, Phys. Lett. B 120:127-32 (1983)

  • L.F. Abbott, P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys.Lett. B120:133-36 (1983)

  • Michael Dine, Willy Fischler, The Not So Harmless Axion, Phys.Lett. B120:137-141 (1983)

A historical recollection of the development until here is in

  • Frank Wilcek, Birth of axions in This Week’s Citation Classic (1991) (pdf)

The argument that the topological interaction term aFF\propto a \langle F \wedge F\rangle gives the axion field aa a vanishing vacuum expectation value is due to

A reformulation of this effect in terms of Chern-Simons forms is discussed in

Review includes:

  • Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), Axions: Theory, cosmology, and Experimental Searches, Lect. Notes Phys. 741 (Springer, Berlin Heidelberg 2008) (doi:10.1007/978-3-540-73518-2_2)

  • Jihn E. Kim, Gianpaolo Carosi, Axions and the Strong CP Problem, Rev.Mod.Phys.82:557-602,2010 (arXiv:0807.3125)

  • Masahiro Kawasaki, Kazunori Nakayama, Axions : Theory and Cosmological Role, Annual Review of Nuclear and Particle Science Vol.63:1-552 (arXiv:1301.1123)

  • Wikipedia Axion

In string theory

Discussion of the various ways that axions naturally appear in string theory is in

Specifically for the F-theory sector of string theory:

Specifically open string axions

  • Gabriele Honecker, section 4 of From Type II string theory towards BSM/dark sector physics, International Journal of Modern Physics A Vol. 31 (2016) 1630050 (arXiv:1610.00007)

A textbook account of axion string phenomenology is in

Discussion of stringy axion cosmology (such as fuzzy dark matter) is in

Experimental signature

Discussion of experimental signatures of and constraints on axion physics:

In particle physics

Discussion of experimental signatures of and constraints on axion physics from particle physics:

Specifically contribution of axions to the anomalous magnetic moment of the electron and muon in QED:

  • Yannis Semertzidis, Magnetic and Electric Dipole Moments in Storage Rings, chapter 6 of Markus Kuster, Georg Raffelt, Berta Beltrán (eds.), Axions: Theory, cosmology, and Experimental Searches, Lect. Notes Phys. 741 (Springer, Berlin Heidelberg 2008) (Kuster-Raffelt-Beltran 08, doi:10.1007/978-3-540-73518-2_2)

  • Roberta Armillis, Claudio Coriano’, Marco Guzzi, Simone Morelli, Axions and Anomaly-Mediated Interactions: The Green-Schwarz and Wess-Zumino Vertices at Higher Orders and g-2 of the muon, JHEP 0810:034,2008 (arXiv:0808.1882)

  • W.J. Marciano, A. Masiero, P. Paradisi, M. Passera, Contributions of axion-like particles to lepton dipole moments, Phys. Rev. D 94, 115033 (2016) (arXiv:1607.01022)

  • Martin Bauer, Matthias Neubert, Andrea Thamm, Collider Probes of Axion-Like Particles, J. High Energ. Phys. (2017) 2017: 44. (arXiv:1708.00443, doi:10.1007/JHEP12(2017)044)

The basic relevant Feynman diagrams are worked out here:

See also

For the ABRACADABRA (A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus) approach to axion detection, see

  • Jonathan Ouellet, ABRACADABRA: A Broadband Search for Axion Dark Matter, talk 2017 (pdf)

For the ADMX experiment:

In astrophysics

Axion signatures in gravitational waves potentially visible by LIGO-type experiments are discussed in

  • Junwu Huang, Matthew C. Johnson, Laura Sagunski, Mairi Sakellariadou, Jun Zhang, Prospects for axion searches with Advanced LIGO through binary mergers (arXiv:1807.02133)

In cosmology

Discussion of experimental signatures of and constraints on axion physics in cosmology, where the axion is a (fuzzy) dark matter-candidate:

  • Joseph P. Conlon, M.C. David Marsh, Searching for a 0.1-1 keV Cosmic Axion Background (arXiv:1305.3603)

    Primordial decays of string theory moduli at z10 12z \sim 10^{12} naturally generate a dark radiation Cosmic Axion Background (CAB) with 0.11keV0.1 - 1 keV energies. This CAB can be detected through axion-photon conversion in astrophysical magnetic fields to give quasi-thermal excesses in the extreme ultraviolet and soft X-ray bands. Substantial and observable luminosities may be generated even for axion-photon couplings 10 11GeV 1\ll 10^{-11} GeV^{-1}. We propose that axion-photon conversion may explain the observed excess emission of soft X-rays from galaxy clusters, and may also contribute to the diffuse unresolved cosmic X-ray background. We list a number of correlated predictions of the scenario.

Discussion of the axion as a candidate for dark matter (fuzzy dark matter) is in

with its experimental bounds

  • Nilanjan Banik, Adam J. Christopherson, Pierre Sikivie, Elisa Maria Todarello, New astrophysical bounds on ultralight axionlike particles, Phys. Rev. D 95, 043542 (2017) (arXiv:1701.04573)

and as a candidate for dark energy:

Last revised on July 9, 2018 at 04:13:39. See the history of this page for a list of all contributions to it.