Charged Gravitational Instantons: Extra CP Violation and Charge Quantisation in the Standard Model

Eur. Phys. J. C (2019) 79:49 https://doi.org/10.1140/epjc/s10052-019-6577-y

Regular Article - Theoretical Physics

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

Suntharan Arunasalama, Archil Kobakhidzeb ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney, NSW 2006 Sydney, Australia

Received: 13 August 2018 / Accepted: 8 January 2019 © The Author(s) 2019

Abstract We argue that quantum electrodynamics com- Furthermore, while for AE manifolds, SEH = 0 implies bined with quantum gravity results in a new source of CP that they are Riemann-flat (no gravity), ALE manifolds with violation, anomalous non-conservation of chiral charge and SEH = 0 necessary have (anti)self-dual Riemann curvature quantisation of electric charge. Further phenomenological tensor. Hence, it is reasonable to think that ALE vacuum man- and cosmological implications of this observation are briefly ifolds describe a topologically non-trivial vacuum structure discussed within the standard model of particle physics and of quantum gravity in close analogy to the instanton vac- cosmology. uum structure in a YangÐMills theory. However, unlike the YangÐMills instanton background, the background of gravitational (anti)self-dual instantons do not support renormal- 1 Introduction izable fermion zero modes. This implies that ALE gravitational instantons do not induce e.g. anomalous violation of Gravitational interactions are typically neglected in parti- a global axial charge and are believed have no phenomeno- cle physics processes, because their local manifestations are logical implications in particle physics.1 minuscule for all practical purposes. However, local phys- The conclusion is dramatically different once one includes ical phenomena are also prescribed by global topological into consideration the Standard Model gauge interactions properties of the theory. In this paper we argue that non- alongside gravity. Namely, we will argue that electrically perturbative quantum gravity effects driven by electrically charged gravitational instantons support fermion zero modes charged gravitational instantons give rise to a topologically and hence induce anomalous chiral symmetry breaking in non-trivial vacuum structure. This in turn leads to important QED. Furthermore, the transition between topologically phenomenological consequences Ð violation of CP symmetry inequivalent vacua mediated by such instantons give rise to and quantisation of electric charge in the standard quantum a θ-vacuum and the CP violation in QED. Gravitationally electrodynamics (QED) augmented by quantum gravity. induced CP violation was first suggested in [5] in the con- Within the Euclidean path integral formalism, quantum text of generic gravitational instantons. We will also argue gravitational effects result from integrating over metric man- that in the background of ALE manifolds that admit spinors, ifolds (M, gμν) with all possible topologies. The definition electric charge is necessarily quantised. In addition, charged of Euclidean path integral for gravity, however, is known gravitational instantons may have important ramifications for to be plagued with difficulties. In particular, the Euclidean cosmology, as it will be briefly discussed at the end of the Einstein-Hilbert action is not positive definite, SEH ≶ 0[1]. paper. Nevertheless, for the purpose of computing quantum gravity contribution to particle physics processes described by flat spacetime S-matrix , we can restrict ourself to asymp- 2 The Eguchi–Hanson instanton totically Euclidean (AE) or asymptotically locally Euclidean (ALE) manifolds. The AE and ALE vacuum manifolds are We start by recalling basic properties of the simplest anti- known to be Ricci flat, R = 0, and have non-negative action, selfdual gravitational instanton, the EguchiÐHanson (EH) SEH 0, according to the positive action theorem [2,3]. 1 It has been suggested that for global gravitational anomalies the rel- a e-mail: [email protected] evant instantons are exotic spheres [4]. The (non)existence of exotic b e-mail: [email protected] spheres in 4D, however, has not been proven yet.

0123456789().: V,-vol 123 49 Page 2 of 5 Eur. Phys. J. C (2019) 79:49 instanton [6].2 The metric for the EH instanton is: at u = 0, one must restrict the domain of ψ to [0, 2π). With this modification, it is clear that the topology of this space 1 a4 ds2 = dr2 + r 2 σ 2 + σ 2 + 1 − σ 2 (1) near the horizon, r = a is that of S2 ×R2 where the sphere is − a4 x y r 4 z 1 r4 parametrised by (θ, φ) and the plane by (u,ψ).Asr →∞, where σi are the differential one-forms: the metric asymptotically approaches that of flat spacetime but with the restriction in the domain of ψ, we see that the 1 boundary at infinity is in fact S3/Z = RP3. σ = (xdt−tdx+ydz−zdy) 2 x r 2 This metric supports a self-dual U(1) gauge field (e.g., the 1 electromagnetic field) of the form [6]: = (sin ψ dθ − sin θ cos ψ dφ) (2) 2 1 A = Aθ = 0 σ = (ydt−tdy+zdx−xdz) r y 2 r qa2 1 Aφ = cos θ = (− cos ψ dθ − sin θ sin ψ dφ) (3) r 2 2 qa2 1 1 Aψ = σz = (zdt−tdz+xdy−ydx) = (dψ + cos θ dφ). 2 r 2 2 r (4) where q is the U(1) charge of the instanton. This instanton satisfies the property: Defining the curvature 2-form as3: μν ˜ μν 1 F = F = √ μνρσ Fρσ . (7) a 1 a μ ν 2 g R = R μν dx ∧ dx , b 2 b It should be noted that no such U(1) instanton solution exists it has been shown in [6] that this metric satisfies the anti- in flat space-time and hence the existence of such charged selfduality property: EguchiÐHanson (CEH) instanton has important phenomenological consequences as seen below. The action of the CEH a 1 c R =− abcd R (5) instanton is given by: b 2 d ≡ a = √ which, in turn, implies that the metric is Ricci flat, R R a 1 4 μν 1 4 SCEH = d x gF Fμν = d xμνρσ Fμν Fρσ 0. 4e2 8e2 The metric is evidently singular at r = a however it can be 4π2q2 = . (8) removedbyperformingaZ2 identification of the coordinates. e2 This can be seen as follows. Letting u = r 1 − a4/r 4,it can be shown that near r = a (or u = 0), the metric can be Note the 4π 2 factor in Eq. (8) vs the standard 8π 2 which rewritten in terms of the Euler angles on S3 as [7]: appears in the action for pure YangÐMills instantons. This is due identification of antipodal points in the EH space, which 1 1 ds2 = du2 + u2(dψ + cos θ dφ)2 becomes a half of the (asymptotic) Euclidean space. More 4 4 importantly, non-perturbative processes are dominated by a2 small-size CEH instantons due to the growing fine structure + (dθ 2 + sin2 θ dφ2). (6) 4 constant α = e2/4π at small scales, in contrast to the dom- inance of large-size instantons in asymptotically free YangÐ θ φ Here, it is evident that at fixed and , the metric becomes the Mills theories. usual metric of a plane with radial coordinate u and angular coordinate ψ. Therefore, to remove the apparent singularity 3 Fermions and their charge quantisation 2 Multi-instanton generation of EH instanton solution is given in Ref. [8]. For a comprehensive review of gravitational instantons, see Ref. [9]. In this section we ask the question whether the CEH space 3 Here and in what follows Greek indices are for curved Euclead- actually admits the existence of fermion fields (spin struc- ean space, while Latin indices are for tangent (Eucleadean flat) space. ture). Consider a path at r =∞from ψ = 0toψ = 2π at γ μ = μγ a σ = i [γ ,γ ] 3 Hence, ea are curved space gamma-matrices, ab 4 a b fixed θ and ϕ. Since the EH space approaches S /Z2 topo- are generators of Eucleadean Lorentz rotations, forming SO(4) symme- →∞ ab logically as r , this path is indeed a loop. Further- try group, and ωμ are spin-connection vector fields of the gauged SO(4) symmetry. As usual, spin-connection fields are expressed through tetrad more, this is loop can not be contracted to a point. This a fields eμ by fulfilling the torsion-free condition. The standard metric can be most easily seen near r = a wherein the space a b 2 2 formulation then is obtained through the relation: gμν = ηabeμeν . approaches R × S with the sphere parametrised by θ and 123 Eur. Phys. J. C (2019) 79:49 Page 3 of 5 49

√ φ and the plane parametrised in plane polar coordinates by / 2 1 μν 1 μν D = √ Dμ gg Dν − iqe Fμνσ (11) u = r 1 − a4/r 4 and ψ and the loop goes encloses the g 2 origin of the plane. We see that the origin in this plane cor- where Dμ =∇μ−iq Aμ. Here, this last operator is indefinite = e responds to the singularity at r a, thus implying that the and hence the prior argument fails to hold. In order to illus- loop is not contractible. Considering a fermion with charge trate the asymptotic behaviour of this zero mode as r →∞, qe moving along such a loop, the additional phase obtained consider the following tetrad frame: is given by: a a μ a μ a e = δ + g−(r)x x + g+(r)x˜ x˜ ∞ 2π μ μ μ = ψ qe Aμdx qe dr d Frψ a 0 ± 1 a4 2 μ ∞ 2 where g±(r) =−1+ 1 − and x˜ = (y, −x, t, −z). −qa r4 = πq dr 4 e 3 It is apparent that these tetrads behave asymptotically as a r xμx δμ + O(a4/r 4) a and hence the spin connections, which =−2πq q. a r2 e are proportional to the derivative of the tetrads also go as O(a4/r 4). In contrast, the U(1) field goes as O(a2/r 2). This accumulated phase must be unobservable for fermion Hence, we can ignore gravitational effects for r >> a and field to be defined consistently [10]. Hence, we require ( ) 4 only consider the U 1 field. In this region, it can be shown qeq = n for some n ∈ Z. Since the smallest observed μ ν that ψ ≈ Fμνγ γ ξ where ξ is a constant spinor is a solu- charge carried by down-type quarks is |q |= 1 (in units of 0 0 e 3 tion of the Dirac equation. This follows from the fact that electron charge), we see that the possible instanton charges μν μ ∂μ F = 0 and the observation that γ Aμψ ∼ O(a4/r 4) are restricted to q = 3n. We find it quite remarkable that which are considered small in this approximation. This solu- the very existence of fermions in quantum gravity combined tion goes as 1/r 4 and hence is normalisable (small-size with electromagnetism automatically implies quantisation of instantons). electric charge. According to the index theorems [11Ð16], the existence of fermion zero modes implies anomalous non-conservation of a chiral charge in gravity-QED mediated nonperturbative 4 Anomalous non-conservation of chiral charge and CP processes. The anomalous divergence of the chiral current violation μ = ψγ¯ γ μψ ψ J5 5 of a fermion carrying a charge qe reads: 2 The EH metric by itself forbids the existence of normalisable μ qe ˜ μν 1 ab ãbμν ∇μ J = Fμν F − R μν R . (12) fermion zero modes even if these are massless. This readily 5 8π 2 192π 2 ∇≡/ γ μ(∂ + can be seen by squaring the Dirac operator i μ By integrating this equation and using Eq. (8) we compute 1 ω abσ ) 4 μ ab : ≡ 3 0 change in chiral charge Q5 d xJ5 due to the CEH 2 μ 1 instanton (the first term on the rhs of 12), ∇/ =∇μ∇ − R. (9) 4 = 2 2 , Q5 2qe q (13) As the EH metric is self-dual, the Ricci scalar, R, is zero. Therefore, given a zero mode ψ, it also satisfies ∇/ 2ψ = while the pure gravitational EH instanton (the second term μ on the the rhs of 12) does not contribute. This non- ∇μ∇ ψ = 0. Then, using partial integration, one finds: conservation comes in addition to the familiar chiral charge d4xψ†∇2ψ =− d4x|∇ψ|2 = 0(10)non-conservation due to the QCD instantons, which is believed to be the origin of the η meson mass [18]. Thus where the anti-hermitian property of the covariant deriva- we expect that CEH instantons also contribute to the low tive was used. Indeed, Eq. (10)impliesthat∇μψ = 0 ⇒ energy QCD physics and it would be interesting to study the ab [∇μ, ∇ν]ψ = 0 ⇒ Rμν σabψ = 0 ⇒ ψ = 0. Hence, related phenomenology. The implications of CEH instantons there are no non-trivial normalisable fermion zero modes in for axion physics is discussed in [19] and some other inter- μ the EH space alone. In other words, −∇μ∇ is a positive esting phenomenological aspects can be found in [20]. definite operator and has no zero modes. Another important phenomenological consequence of In contrast, however, if one also introduces the U(1) gauge CEH instantons is the violation of CP parity in QED and field, the picture changes significantly. The squared equation gravity. Since CEH instantons mediate transitions between operator now becomes: topologically distinct vacuum states, the standard cluster decomposition argument implies the following CP violat- 4 This is reminiscent of the Dirac quantisation condition in the presence ing topological terms are necessarily present in the effective of magnetic monopoles [17]. Lagrangian: 123 49 Page 4 of 5 Eur. Phys. J. C (2019) 79:49

1 ˜ μν ab ˜abμν and cosmology, which has not been fully appreciated previ- √ LCP/ = θQEDFμν F + θgrav R μν R . (14) g ously.

The ﬁrst term is supported by CEH instantons and the sec- Acknowledgements The authors are indebted to Zurab Berezhiani, ond term exists even if q = 0. Note that in the presence Gerard ’t Hooft and Arkady Vainshtein for stimulating discussions and Stanley Deser and Michael Duff for their email correspondence. The of massless charged fermions θQED becomes unphysical work was partially supported by the Australian Research Council. AK is θ v and can be removed by chiral transformations, while gra also indebted to organisers of the INT Workshop “Neutron-Antineutron remains, since vacuum-to-vacuum transitions due to the pure Oscillations: Appearance, Disappearance, and Baryogenesis” for the EH instantons remain effective due to the absence of fermion opportunity to present preliminary results of this research. zero modes even for massless fermions. The potential con- Data Availability Statement This manuscript has no associated data tribution of the gravitationally induced CP-violating observ- or the data will not be deposited. [Author’s comment: Data sharing is ables such as the electric dipole moments of charged particles not applicable to this article as no datasets were generated or analysed are currently under study [21]. during the current study.]

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm 5 Outlook: embedding into the Standard Model ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative The phenomenological implications of CEH instantons Commons license, and indicate if changes were made. become even richer when one considers the full Standard Funded by SCOAP3. Model [21]. The EH instantons charged under the electroweak group SU(2) × U(1) lead to an anomalous violation of the lepton number in the Standard Model (assum- References ing the absence of right-handed sterile neutrinos) and induce new electroweak CP phases associated with the weak isospin 1. G.W. Gibbons, S.W. Hawking, M.J. Perry, Nucl. Phys. B 138, 141 and hypercharge groups. This may have several interest- (1978). https://doi.org/10.1016/0550-3213(78)90161-X ing ramifications which deserve further study. In particu- 2. R.M. Schon, S.T. Yau, Phys. Rev. Lett. 42, 547 (1979). https://doi. lar, the electroweak CEH instantons may generate nonper- org/10.1103/PhysRevLett.42.547 3. E. Witten, Commun. Math. Phys. 80, 381 (1981). https://doi.org/ turbative masses for neutrinos, providing instantonic reali- 10.1007/BF01208277 sation of the gravitational neutrino mass generation mecha- 4. E. Witten, Commun. Math. Phys. 100, 197 (1985). https://doi.org/ nism recently suggested in [22]. The simultaneous break- 10.1007/BF01212448 ing of CP and lepton number by gravitational instantons 5. S. Deser, M.J. Duff, C.J. Isham, Phys. Lett. 93B, 419 (1980). https:// doi.org/10.1016/0370-2693(80)90356-1 could be the source of the observed baryon number (B) 6. T. Eguchi, A.J. Hanson, Phys. Lett. 74B, 249 (1978). https://doi. asymmetry in the universe. The scenario we keep in mind org/10.1016/0370-2693(78)90566-X is that the electroweak CEH instantons (or perhaps equiv- 7. T. Eguchi, A.J. Hanson, Ann. Phys. 120, 82 (1979). https://doi.org/ alent sphalerons), through nontrivial field configurations, 10.1016/0003-4916(79)90282-3 8. G.W. Gibbons, S.W. Hawking, Phys. Lett. 78B, 430 (1978). https:// induce a lepton number (L) asymmetry at high temper- doi.org/10.1016/0370-2693(78)90478-1 atures. The required departure from thermal equilibrium 9. T. Eguchi, P.B. Gilkey, A.J. Hanson, Phys. Rep. 66, 213 (1980). would be automatically guaranteed, since that gravitational https://doi.org/10.1016/0370-1573(80)90130-1 interactions below the Planck scale cannot sustain in ther- 10. S.W. Hawking, C.N. Pope, Phys. Lett. 73B, 42 (1978). https://doi. org/10.1016/0370-2693(78)90167-3 mal equilibrium. The generated lepton asymmetry is then 11. M.F. Atiyah, I.M. Singer, Bull. Am. Math. Soc. 69, 422 (1969). partly transferred into baryon asymmetry due to the equi- https://doi.org/10.1090/S0002-9904-1963-10957-X librium B+L number violating processes induced by elec- 12. M.F. Atiyah, I.M. Singer, Ann. Math. 87, 484 (1968). https://doi. troweak sphalerons. org/10.2307/1970715 13. M.F. Atiyah, I.M. Singer, Ann. Math. 87, 546 (1968). https://doi. To conclude, we have argued that charged EguchiÐHanson org/10.2307/1970717 gravitational instantons would have a number of impor- 14. M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. tant implications for particle physics. Namely, we have Soc. 77, 43 (1975). https://doi.org/10.1017/S0305004100049410 identified new CP violating phases and chiral and lep- 15. M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. Soc. 78, 405 (1976). https://doi.org/10.1017/S0305004100051872 ton number violating non-perturbative processes associated 16. M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. with CEH instantons within the Standard Model, without Soc. 79, 71 (1976). https://doi.org/10.1017/S0305004100052105 extending its particle content. It also provides a theoret- 17. P.A.M. Dirac, Proc. R. Soc. Lond. A 133, 60 (1931). https://doi. ical explanation of the observed quantisation of electric org/10.1098/rspa.1931.0130 18. G. ’t Hooft, Phys. Rev. D 14 (1976) 3432 Erratum: [Phys. Rev. charge of elementary fermions. All this points towards a D 18 (1978) 2199]. https://doi.org/10.1103/PhysRevD.18.2199.3, rather prominent role of quantum gravity in particle physics https://doi.org/10.1103/PhysRevD.14.3432 123 Eur. Phys. J. C (2019) 79:49 Page 5 of 5 49

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