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CPT-violating leptogenesis induced by gravitational defects

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Abstract

We explore leptogenesis induced by the propagation of neutrinos in gravitational backgrounds that may occur in string theory. The first background is due to linear dilatons and the associated Kalb–Ramond field (axion) in four non-compact space–time dimensions of the string, and can be described within the framework of local effective Lagrangians. The axion is linear in the time coordinate of the Einstein frame and gives rise to a constant torsion which couples to the fermion spin through a gravitational covariant derivative. This leads to different energy-momentum dispersion relations for fermions and antifermions. As a result leptogenesis and baryogenesis can arise in various scenarios. The next two backgrounds go beyond the local effective Lagrangian framework. One is a stochastic (Lorentz violating) Finsler metric which again leads to different dispersion relations between fermions and antifermions. The third background of primary interest is the one due to populations of stochastically fluctuating point-like space–time defects that can be encountered in string/brane theory (D0-branes). Only neutral matter interacts non-trivially with these intrinsic defects, as a consequence of charge conservation. Hence, such a background singles out neutrinos among the Standard Model excitations as the ones interacting predominantly with the defects. The back-reaction of the defects on the space–time due to their interaction with neutral matter results in stochastic Finsler-like metrics (similar to our second background). On average, the stochastic fluctuations of the D0-brane defects preserve Lorentz symmetry, but their variance is nonzero. Interestingly, the particle–antiparticle asymmetry comes out naturally to favour matter over antimatter in this third background, once the effects of the kinematics of the scattering of the D-branes with matter is incorporated.

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Notes

  1. The above considerations concern the dispersion relations for any fermion, not only neutrinos. However, when one considers matter excitations from the vacuum, as relevant for leptogenesis, we need chiral fermions to get non-trivial CPTV asymmetries in populations of particle and antiparticles, because \(\langle\psi^{\dagger}\gamma^{5} \psi\rangle = - \langle\psi_{L}^{\dagger}\gamma^{5} \psi_{L}\rangle + \langle\psi_{R}^{\dagger}\gamma^{5} \psi_{R}\rangle \).

  2. We note that fermions coupled to Kalb–Ramond torsion tensors H μνρ have been considered previously [50, 51] but from a different perspective than ours.

  3. We mention, for completeness, that the other popular class of Finsler geometries, which appears in the general relativistic version [63–66] of the so-called Very Special Relativity Model [78], and cosmological extensions thereof [71, 73], are not characterised by CPTV in the dispersion relations of the spin-curvature type discussed in Sect. 2.3. In fact such VSR-related models have been proposed in the past as candidates for the generation of L conserving neutrino masses [79], and hence our L violating considerations in this work do not apply.

  4. It should be remarked that for the effective compactified D-“particles” the interactions with the charged matter excitations are suppressed relative to the neutral ones [82]. Hence, even in this case, it is the electrically neutral excitations which interact primarily with the D-foam.

  5. For brevity, in what follows we ignore potential contributions induced by compactification of the D8 brane worlds to D3 branes, stating only the expressions for the induced potential on the uncompactified brane world as a result of a stretched string between the latter and the D-particle—the compactification does not affect our arguments on the negative energy contributions to the brane vacuum energy.

  6. Ignoring the flavour structure, the metric (53) can be written as

    $$ ds^2 = dt^2 + 2 u_i\, dx^i\, dt - \delta_{ij}\, dx^i\, dx^j . $$
    (55)

    This metric was determined from world-sheet conformal field theory considerations [76] and represents a dragging of the frame by the Galilean (slowly moving) D particle, which moves on a flat space–time background. However, the string excitations represent relativistic particles, and as such they move according to the rules of special relativity. Any four vectors attached to the strings, such as a four velocity, will evolve by a series of infinitesimal Lorentz boosts induced by the change of the D particle velocity relative to the particle. In this sense, one may perform a time coordinate change in the metric (55) to write in the form, up to terms u 3 for small recoil velocities |u|≪1,

    $$ ds^2 = dt_\mathrm{ff} ^2 + 2 u_i\, dx^i\, dt_\mathrm{ff} - \delta_{ij} \bigl(dx^i - u^i\, dt_\mathrm{ff} \bigr) \bigl(dx^j - u^j\, dt_\mathrm{ff}\bigr) + \mathcal{O} \bigl(u^3\bigr). $$
    (56)

    The metric (56) is nothing but the so-called Gullstrand–Painlevé metric [89], representing the geometry in the exterior of a Schwarzschild black hole, where the falling space into the black hole is represented as a Galilean river on a flat space–time in which relativistic fishes swim. The river represents the frame of the recoiling D particle, while the fishes are the relativistic matter strings. Here t ff is the time of a free-floating observer who is at rest at infinity (compared to the centre of the black hole). In the case of a black hole the relative velocities u i are coordinate dependent, of course, unlike our approximation in the D-foam case, although one may easily consider more general cases, where the recoil velocities of the D-particles in the foam are non-uniform, in which case the analogy with the Gullstrand–Painlevé river would become stronger.

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Acknowledgements

The work of N.E.M. was supported in part by the London Centre for Terauniverse Studies (LCTS), using funding from the European Research Council via the Advanced Investigator Grant 267352. N.E.M. and S.S. also thank the STFC UK for partial support under the research grant ST/J002798/1.

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    Present address: Theory Division, Physics Department, CERN, 1211, Geneva 23, Switzerland

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  1. Theoretical Particle Physics and Cosmology Group, Department of Physics, King’s College London, Strand, London, WC2R 2LS, UK

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Mavromatos, N.E., Sarkar, S. CPT-violating leptogenesis induced by gravitational defects. Eur. Phys. J. C 73, 2359 (2013). https://doi.org/10.1140/epjc/s10052-013-2359-0

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