# Charged gravitational instantons: extra CP violation and charge quantisation in the Standard Model

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## Abstract

We argue that quantum electrodynamics combined with quantum gravity results in a new source of CP violation, anomalous non-conservation of chiral charge and quantisation of electric charge. Further phenomenological and cosmological implications of this observation are briefly discussed within the standard model of particle physics and cosmology.

## 1 Introduction

Gravitational interactions are typically neglected in particle physics processes, because their local manifestations are minuscule for all practical purposes. However, local physical phenomena are also prescribed by global topological properties of the theory. In this paper we argue that non-perturbative quantum gravity effects driven by electrically charged gravitational instantons give rise to a topologically non-trivial vacuum structure. This in turn leads to important phenomenological consequences – violation of CP symmetry and quantisation of electric charge in the standard quantum electrodynamics (QED) augmented by quantum gravity.

Within the Euclidean path integral formalism, quantum gravitational effects result from integrating over metric manifolds \((M, g_{\mu \nu })\) with all possible topologies. The definition of Euclidean path integral for gravity, however, is known to be plagued with difficulties. In particular, the Euclidean Einstein-Hilbert action is not positive definite, \(S_{EH}\lessgtr 0\) [1]. Nevertheless, for the purpose of computing quantum gravity contribution to particle physics processes described by flat spacetime *S*-matrix , we can restrict ourself to asymptotically Euclidean (AE) or asymptotically locally Euclidean (ALE) manifolds. The AE and ALE vacuum manifolds are known to be Ricci flat, \(R=0\), and have non-negative action, \(S_{EH}\geqslant 0\), according to the positive action theorem [2, 3]. Furthermore, while for AE manifolds, \(S_{EH}=0\) implies that they are Riemann-flat (no gravity), ALE manifolds with \(S_{EH}=0\) necessary have (anti)self-dual Riemann curvature tensor. Hence, it is reasonable to think that ALE vacuum manifolds describe a topologically non-trivial vacuum structure of quantum gravity in close analogy to the instanton vacuum structure in a Yang–Mills theory. However, unlike the Yang–Mills instanton background, the background of gravitational (anti)self-dual instantons do not support renormalizable fermion zero modes. This implies that ALE gravitational instantons do not induce e.g. anomalous violation of a global axial charge and are believed have no phenomenological implications in particle physics.^{1}

The conclusion is dramatically different once one includes into consideration the Standard Model gauge interactions alongside gravity. Namely, we will argue that electrically charged gravitational instantons support fermion zero modes and hence induce anomalous chiral symmetry breaking in QED. Furthermore, the transition between topologically inequivalent vacua mediated by such instantons give rise to a \(\theta \)-vacuum and the CP violation in QED. Gravitationally induced *CP* violation was first suggested in [5] in the context of generic gravitational instantons. We will also argue that in the background of ALE manifolds that admit spinors, electric charge is necessarily quantised. In addition, charged gravitational instantons may have important ramifications for cosmology, as it will be briefly discussed at the end of the paper.

## 2 The Eguchi–Hanson instanton

^{2}The metric for the EH instanton is:

^{3}:

*u*and angular coordinate \(\psi \). Therefore, to remove the apparent singularity at \(u=0\), one must restrict the domain of \(\psi \) to \([0,2\pi )\). With this modification, it is clear that the topology of this space near the horizon, \(r=a\) is that of \(S^2\times {\mathbb {R}}^2\) where the sphere is parametrised by \((\theta ,\phi )\) and the plane by \((u, \psi )\). As \(r\rightarrow \infty \), the metric asymptotically approaches that of flat spacetime but with the restriction in the domain of \(\psi \), we see that the boundary at infinity is in fact \(S^3/{\mathbb {Z}}_2={\mathbb {R}}P^3\).

*U*(1) gauge field (e.g., the electromagnetic field) of the form [6]:

*U*(1) charge of the instanton. This instanton satisfies the property:

*U*(1) instanton solution exists in flat space-time and hence the existence of such charged Eguchi–Hanson (CEH) instanton has important phenomenological consequences as seen below. The action of the CEH instanton is given by:

## 3 Fermions and their charge quantisation

^{4}Since the smallest observed charge carried by down-type quarks is \(|q_e|=\frac{1}{3}\) (in units of electron charge), we see that the possible instanton charges are restricted to \(q=3n\). We find it quite remarkable that the very existence of fermions in quantum gravity combined with electromagnetism automatically implies quantisation of electric charge.

## 4 Anomalous non-conservation of chiral charge and CP violation

*R*, is zero. Therefore, given a zero mode \(\psi \), it also satisfies Open image in new window . Then, using partial integration, one finds:

*U*(1) gauge field, the picture changes significantly. The squared equation operator now becomes:where \(D_\mu =\nabla _\mu -i q_e A_\mu \). Here, this last operator is indefinite and hence the prior argument fails to hold. In order to illustrate the asymptotic behaviour of this zero mode as \(r\rightarrow \infty \), consider the following tetrad frame:

*U*(1) field goes as \(O(a^2/r^2)\). Hence, we can ignore gravitational effects for \(r>> a\) and only consider the

*U*(1) field. In this region, it can be shown that \(\psi \approx F_{\mu \nu }\gamma ^\mu \gamma ^\nu \xi _0\) where \(\xi _0\) is a constant spinor is a solution of the Dirac equation. This follows from the fact that \(\partial _\mu F^{\mu \nu }=0\) and the observation that \(\gamma ^\mu A_\mu \psi \sim O(a^4/r^4)\) which are considered small in this approximation. This solution goes as \(1/r^4\) and hence is normalisable (small-size instantons).

## 5 Outlook: embedding into the Standard Model

The phenomenological implications of CEH instantons become even richer when one considers the full Standard Model [21]. The EH instantons charged under the electroweak group \(SU(2)\times U(1)\) lead to an anomalous violation of the lepton number in the Standard Model (assuming the absence of right-handed sterile neutrinos) and induce new electroweak CP phases associated with the weak isospin and hypercharge groups. This may have several interesting ramifications which deserve further study. In particular, the electroweak CEH instantons may generate nonperturbative masses for neutrinos, providing instantonic realisation of the gravitational neutrino mass generation mechanism recently suggested in [22]. The simultaneous breaking of CP and lepton number by gravitational instantons could be the source of the observed baryon number (B) asymmetry in the universe. The scenario we keep in mind is that the electroweak CEH instantons (or perhaps equivalent sphalerons), through nontrivial field configurations, induce a lepton number (L) asymmetry at high temperatures. The required departure from thermal equilibrium would be automatically guaranteed, since that gravitational interactions below the Planck scale cannot sustain in thermal equilibrium. The generated lepton asymmetry is then partly transferred into baryon asymmetry due to the equilibrium B+L number violating processes induced by electroweak sphalerons.

To conclude, we have argued that charged Eguchi–Hanson gravitational instantons would have a number of important implications for particle physics. Namely, we have identified new CP violating phases and chiral and lepton number violating non-perturbative processes associated with CEH instantons within the Standard Model, without extending its particle content. It also provides a theoretical explanation of the observed quantisation of electric charge of elementary fermions. All this points towards a rather prominent role of quantum gravity in particle physics and cosmology, which has not been fully appreciated previously.

## Footnotes

- 1.
It has been suggested that for global gravitational anomalies the relevant instantons are exotic spheres [4]. The (non)existence of exotic spheres in 4D, however, has not been proven yet.

- 2.
- 3.
Here and in what follows Greek indices are for curved Eucleadean space, while Latin indices are for tangent (Eucleadean flat) space. Hence, \(\gamma ^{\mu }=\mathrm{e}^{\mu }_a\gamma ^{a}\) are curved space gamma-matrices, \(\sigma _{ab}=\frac{i}{4}[\gamma _a,\gamma _b]\) are generators of Eucleadean Lorentz rotations, forming SO(4) symmetry group, and \(\omega _{\mu }^{~ab}\) are spin-connection vector fields of the gauged SO(4) symmetry. As usual, spin-connection fields are expressed through tetrad fields \(\mathrm{e}^{a}_{\mu }\) by fulfilling the torsion-free condition. The standard metric formulation then is obtained through the relation: \(g_{\mu \nu }=\eta _{ab}\mathrm{e}^{a}_{\mu }\mathrm{e}^{b}_{\nu }\).

- 4.
This is reminiscent of the Dirac quantisation condition in the presence of magnetic monopoles [17].

## Notes

### Acknowledgements

The authors are indebted to Zurab Berezhiani, Gerard ’t Hooft and Arkady Vainshtein for stimulating discussions and Stanley Deser and Michael Duff for their email correspondence. The work was partially supported by the Australian Research Council. AK is also indebted to organisers of the INT Workshop “Neutron-Antineutron Oscillations: Appearance, Disappearance, and Baryogenesis” for the opportunity to present preliminary results of this research.

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