Gravitational leptogenesis, C, CP and strong equivalence
The origin of matter-antimatter asymmetry is one of the most important
outstanding problems at the interface of particle physics and
cosmology. Gravitational leptogenesis (baryogenesis) provides a
possible mechanism through explicit couplings of spacetime curvature
to appropriate lepton (or baryon) currents. In this paper, the idea
that these strong equivalence principle violating interactions could
be generated automatically through quantum loop effects in curved
spacetime is explored, focusing on the realisation of the
discrete symmetries C, CP and CPT which must be broken to induce
matter-antimatter asymmetry. The related issue of quantum corrections
to the dispersion relation for neutrino propagation in curved
spacetime is considered within a fully covariant framework.
Swansea, SA2 8PP, UK. \notoc
The origin of matter-antimatter asymmetry in the universe remains one of the outstanding problems at the interface of particle physics and cosmology. In recent years, fresh impetus has been given to this issue by the development of models where gravity plays a key role in the symmetry breaking dynamics.
The motivation for looking at gravitational leptogenesis111In this paper we focus on leptogenesis, although all the theoretical development would transfer immediately to direct models of gravitational baryogenesis. A lepton-antilepton asymmetry may also be transferred to a baryon-antibaryon asymmetry by the standard sphaleron mechanism at non-zero temperature Kuzmin:1985mm (). arises from a critical analysis of the Sakharov conditions Sakharov () for the generation of matter-antimatter asymmetry in cosmology. These require models of lepto/baryogenesis to contain (i) a mechanism for lepton/baryon number violation, (ii) C and CP violation, (iii) non-equilibrium dynamics. Subsequently, it was realised that in a gravitational field, the final criterion may be replaced by an effective violation of CPT symmetry Cohen:1987vi (). Models with C and CP-violating gravitational couplings introduced by hand in a more or less well-motivated way have also been proposed to address the second Sakharov condition (see e.g. ref.Lambiase:2013haa () for a review). An important example is the interaction , where is the lepton current, introduced in the model of ref.Davoudiasl:2004gf (). Here, the time derivative of the Ricci scalar, , may act as a chemical potential for lepton number, inducing a lepton-antilepton asymmetry at non-zero temperature.
On the other hand, it is known that in quantum field theory in curved spacetime, quantum loop effects induce effective violations of the strong equivalence principle in the sense that the corresponding effective Lagrangian contains interaction terms which depend explicitly on the curvature, such as or Drummond:1979pp (); Ohkuwa:1980jx (). The question naturally arises whether we can use this mechanism of radiatively-induced strong equivalence violation to automatically generate curvature interactions relevant for leptogenesis. Moreover, a better understanding of the mechanism – especially of the role of the discrete symmetries C, P and T and the key combinations CP and CPT – should provide a guide to the properties of BSM models necessary for gravitational leptogenesis to work. The purpose of this paper is to develop a better theoretical understanding of these fundamental issues.
We should be clear to distinguish two interpretations of “symmetry breaking” in this context. First there is the question of whether the full terms arising in the effective Lagrangian, including the curvature, are invariant under the symmetries of the original theory; in particular, whether discrete symmetries present at tree level may be violated by quantum loop effects. For example, we may ask whether an interaction of the form , which is CP odd, can arise from a CP-conserving tree-level action. We refer to this as “anomalous symmetry breaking”. For this, we require a careful discussion of the realisation of discrete symmetries in a spinor theory in curved spacetime.
However, at a given point in spacetime where the background curvature takes a fixed value, the effective Lagrangian resembles the Lorentz and CPT-violating actions proposed by Kostelecky and collaborators Colladay:1998fq (), with the background field curvature playing the role of coupling constants for potentially C, CP or CPT-violating operators. For example, the Lorentz-violating Dirac action contains a term where the coupling constant multiplies an operator which is CPT odd. It follows that in a fixed background, the radiatively-induced curvature interactions may effectively violate these discrete symmetries, giving rise to phenomenological effects comparable to those in explicit Lorentz-violating theories. We will refer to this as “environmental symmetry breaking” to emphasise the distinction.
Although we are primarily motivated by possible applications to leptogenesis, the paper is also concerned with more general theoretical issues related to quantum loop effects in spinor field theories in curved spacetime, especially gravitational effects on neutrino propagation. In fact, these are closely related since any gravity-induced change in the neutrino dispersion relation which distinguishes between left and right-handed fermions would induce a matter-antimatter asymmetry.
The paper is organised as follows. In section 2, we present a fully covariant derivation of the dispersion relation for Dirac fermions propagating in curved spacetime, in the framework of the eikonal approximation. Establishing and understanding this requires a careful treatment of the role of local inertial frames and the spin connection, so we begin with an extended discussion of the geometry of spinors in curved spacetime. This discussion sharpens our critique of a number of proposals in the literature, in particular the suggestion in refs.Singh:2003sp (); Mukhopadhyay:2005gb (); Debnath:2005wk () that leptogenesis can arise simply through the coupling of neutrinos to the spin connection in the tree-level Dirac action.
Section 3 contains our main analysis of the effective violation of the strong equivalence principle by quantum loops in the case of neutrinos in the standard model. We determine the one-loop effective Lagrangian, emphasising the importance of using a complete basis of hermitian operators, by matching coefficients with explicit Feynman diagram calculations in a weak background field. This generalises earlier work on neutrino propagation in curved spacetime by Ohkuwa Ohkuwa:1980jx ().
We then analyse in some detail the discrete symmetry properties of the operators arising in the effective Lagrangian. We verify that in this model, only CP even operators arise, respecting the symmetries of the original tree-level action. There is no evidence, even in curved spacetime, for an anomalous violation of C, CP or CPT symmetry at the quantum loop level. In particular, a CP-violating interaction of the type required by the Davoudiasl et al. model of leptogenesis Davoudiasl:2004gf () does not arise through radiative corrections in the CP-conserving sector of the standard model. We conclude that such curvature interactions would only arise in theories in which some CP violation is already present in the original Lagrangian.
Applications of strong equivalence violating curvature interactions in gravitational leptogenesis generally rely either on identifying an interaction as analogous to a chemical potential for lepton number or inferring a splitting in energy levels for particles and antiparticles through dispersion relations. With this motivation, in section 4 we study in some detail the dispersion relations arising from the operators which appear in the effective Lagrangian of section 3, also including the CP-violating operator described above. The occurrence of a hierarchy of scales when curvature interactions are present requires some generalisation of the eikonal approximation discussed in section 2. This also allows us to determine the quantum loop corrections to the neutrino dispersion relation in the standard model, verifying the result of Ohkuwa:1980jx () that the low-frequency phase velocity for massless neutrinos is superluminal for backgrounds satisfying the null-energy condition.
Finally, in section 5, we summarise our conclusions and discuss the implication of the theoretical issues raised in our work to models which attempt to generate matter-antimatter asymmetry through gravitational leptogenesis.
2 Inertial Frames and Spinors
2.1 Spinor Formalism in Curved Spacetime
In this section we give a brief review of the main elements of the spinor formalism in curved spacetime we need in the paper (see Bir (); Free () for a more complete account). In a general background, Lorentz transformations can only be realised locally in the tangent plane at each spacetime point. This is achieved by introducing an orthonormal basis satisfying
Here, Greek indices label coordinate basis components and Latin indices label the different vierbein basis vectors . Lorentz transformations are defined as any transformation of the vierbein
which preserves the relation (1), where form a basis for the fundamental representation of the Lorentz algebra. Thus the vierbein provides a rectangular frame on which one can perform local boosts and rotations.
Now that we have formulated Lorentz transformations, we can introduce particles in the spin-1/2 representation of the Lorentz group. Since the Lorentz transformations are local, this necessitates the introduction of a gauge connection or spin-connection in the spinor representation of the Lorentz algebra
where . The covariant derivative is then defined by
which transforms in the same way as under SO(1,3) transformations:
provided that the spin connection transforms as
Defining , one can construct a Lorentz invariant Dirac action
One can find a relation between the spin-connection , vierbein and Christoffel symbols in the following way. Consider a vector . We have that
We can also write the derivative in the vierbein basis , which after a little algebra leads to the relation
With regard to notation, we will use to denote the derivative which is both a and tensor. Its action on the vierbein is defined by
so that (12) is equivalent to the condition
We will use to denote the derivative which is a tensor, but not a tensor:
This allows the relation (12) to be written in a more compact form as
In the next section we introduce the concept of a non-accelerating vierbein frame, and use this together with the relation (16) to show how the spin connection must vanish at the origin of such a frame and hence that the Dirac equation satisfies the strong equivalence principle.
2.2 Inertial Frames
In flat space, the Cartesian tetrad has constant components along any curve, and thus defines an inertial tetrad throughout spacetime. In curved space, an “inertial” tetrad can only be defined in the neighbourhood of a specific reference point corresponding to the inertial observer. The covariant generalisation of “constant components” along a curve is parallel transport.
We construct an inertial frame about a given point as follows. Choose any orthonormal tetrad at and define the vierbein in the neighbourhood of by parallel transport of the vierbein along every curve emanating from (see figure 1). It is now easy to see that the spin connection must vanish at . Pick any coordinate chart in the neighbourhood of the point , and let be the tangent vector of any curve through . Then we have
but the parallel transport condition means that222Notice the derivative here gives an SO(1,3) gauge fixing, and should not be confused with the one in the equation , which is SO(1,3) invariant. at and since is arbitrary it follows that vanishes at . The existence of a local inertial frame is guaranteed by the assumption that the spacetime is Riemannian, the mathematical realisation of the weak equivalence principle.
The physical interpretation is that parallel transport ensures the vierbein, which may be thought of as a set of measuring rods, is not accelerating as it approaches . For a given direction specified by a curve through with tangent vector , one can define an acceleration 4-vector for each vierbein component
It is then easy to see that our prescription simply imposes the condition that the 4-acceleration of any measuring rod is zero in all directions approaching . Put another way, it means that only observers whose “measuring rods” are accelerating will measure the spin connection at .
Another way to understand the parallel transport condition is to set up Riemann normal coordinates centered on . In these coordinates and so the condition is simply
In other words, observers with inertial coordinates perceive the inertial vierbein to have constant components in an infinitesimal neighbourhood of . It is now easy to see that for an inertial tetrad the Dirac equation satisfies
at . This satisfies the strong equivalence principle, viz. it reduces to its special relativistic form in an inertial frame. As we have seen, this corresponds to the requirement that the curved space Lagrangian involves only the connection, with no explicit curvature terms. We will see how this is affected by quantum corrections later in the paper.
We should also mention that one can define an inertial vierbein along a curve associated to a geodesic observer with tangent vector by choosing (with the normalisation ) and defining in the neighborhood by parallel transport along spacelike geodesics normal to . This fixes the SO(1,3) gauge in the normal convex neighbourhood of . One can then define an inertial set of coordinates around in which the Christoffel symbols vanish along , i.e. . Physically this corresponds to a freely falling observer carrying a gyroscopic set of measuring rods. The full mathematical formulation of this concept is Fermi normal coordinates as discussed at length in Poisson:2011nh ().
2.3 Particle Propagation and the Dirac Equation
In curved spacetime, familiar Minkowski space concepts such as particle momenta and trajectories, spin states and dispersion relations are no longer directly applicable and their generalisation requires a subtle and careful analysis of the Dirac equation and its relation to the underlying geometry. Our starting point is the Dirac equation in curved spacetime
In flat space, particles are identified with plane waves, but the curved space Dirac equation will not in general admit such solutions. However, in a kinematical regime where the curvature scale is relatively long compared to the wavelength, we can find solutions which locally resemble plane waves and thus exhibit particle-like properties.
To construct these quasi-plane wave solutions, we use an eikonal approach familiar from geometric optics in curved space MTW (); Schneider (). These solutions are characterised by a wavelength where is the wave-vector and is the scale over which the amplitude varies (see figure 2).
Two other scales enter the analysis – the Compton wavelength of the Dirac particle and the curvature scale , where represents the size of a typical curvature tensor component. We need to decide from the outset how these relate to the hierarchy of scales . Since in the flat-space limit we want to recover the standard dispersion relation , we should clearly take . We also identify with , since it is the background gravitational field that determines the scale over which the amplitude of the quasi-plane waves will vary. If we were to take the curvature scale comparable to or , the solutions would no longer resemble plane waves and we would lose any interpretation in terms of particle states carrying definite momenta.
In the eikonal approximation, we split the solutions into a rapidly-varying phase and a slowly-varying amplitude ( = 1,2) multiplying basis spinors (or for the corresponding antiparticle solutions), i.e.
The book-keeping parameter (MTW (); Schneider ()), which is finally set to 1, identifies the order of the associated quantities in powers of the parameter . We then solve the equation order by order in an expansion in powers of . To implement the condition automatically, we should also take the mass in the Dirac equation (21).
The simplest way to derive the key results, and to compare with previous analyses of the Maxwell equation and photon propagation Shore:2003jx (), is to square the Dirac equation and consider the wave equation
Notice that the Ricci scalar arises from the identity
and gamma matrix manipulations. Now insert the eikonal ansatz (22),(23) into (24) and identify as the wave-vector, which is orthogonal to the wavefronts of constant phase . Collecting terms of the same order in , we find a solution by satisfying the following equations sequentially for , and :
where is a shorthand for with no sum on . Equation (26) recovers the expected dispersion relation
To understand the geometric significance of the second equation, consider the congruence defined by the tangent vectors . These curves, which are timelike geodesics, are identified as the particle trajectories in curved space. The geodesic property follows immediately from (29) by taking a covariant derivative, and using the identity , giving the geodesic equation
Defining , we may identify the optical scalars (expansion), (shear) and (twist) of the congruence as
where we define the projection operator for the hypersurfaces of constant phase by
The congruence is twist-free (and therefore surface-forming) by virtue of the fact that is a gradient, so . The divergence measures the rate of expansion of the congruence, measures the tendency of the congruence to twist and the shear corresponds to geodesics moving apart in one direction and together in the orthogonal direction, whilst preserving the cross-sectional area.
Given , it follows that the equation (27) becomes
If we choose a solution in which the basis spinors are paralelly transported:
the equation (36) becomes
which shows that at leading order the amplitude is governed geometrically by the expansion rate of the geodesic congruence. The equation (28) then determines the sub-leading amplitude correction in terms of , and so on.
So far, our results are independent of a particular choice of frame. Now, choose a particular timelike geodesic within the congruence and imagine a co-moving observer freely-falling with the particle along this trajectory, measuring the evolution of its spin polarization. The observer is equipped with an inertial vierbein, where we identify the timelike vierbein component with and demand that the spatial vierbein vectors are parallel transported in the neighbourhood of . In this inertial frame the connection vanishes along the trajectory,
showing that in this inertial frame the basis spinors are constant along the trajectory, while the amplitude satisfies
Several key points need to be emphasised here:
in the eikonal approximation, we recover a particle interpretation even in curved space, with Dirac particles propagating along timelike geodesics;
the connection does not appear in the dispersion relation which, within the eikonal approximation, is identical to its flat space form consistent with the weak and strong equivalence principles;
the spin polarisation is parallel propagated along the trajectory, and is constant viewed in an inertial frame;
the wave amplitude (particle density) is governed by the expansion optical scalar of the associated geodesic congruence;
the curvature only affects the amplitude, not the dispersion relation, and only at higher order in the eikonal approximation as given by (28).
While this reveals the essential physics, and allows an easy comparison with photon propagation, a more rigorous treatment demands that we solve the Dirac equation itself in this framework rather than just the associated wave equation (24). This was first carried out by Audretsch Audretsch:1981wf () and we present a simplified version of his analysis here.
Acting with the Dirac operator on the eikonal ansatz (42) and collecting terms of the same order in as before, this time we find
The first equation is now an algebraic matrix equation and the existence of a non-trivial solution requires
As before we want to choose basis spinors which satisfy the normalisation
and the parallel propagation property
One possible choice is 333It can be most easily checked that these satisfy these two conditions by evaluating in the rest frame of the particle where and using the fact they are SO(1,3) tensor identities.
Taking a covariant derivative of (47) gives
so that expanding (46) as
Finally, we should perhaps emphasise that while this shows that the dispersion relation is independent of the spin connection in the free Dirac Lagrangian, it does not mean that the gravitational field has no influence the dynamics of the fermion’s spin. Indeed, Audretsch Audretsch:1981wf () has analysed the propagation of the fermion current in more detail, using a Gordon decomposition into a ‘convection current’ and a ‘spin magnetisation current’ . At leading order in the eikonal expansion, the convection current follows the timelike geodesic defined by (at higher order there are tidal curvature corrections), while the spin motion is defined by parallel propagation.
2.4 Dispersion Relations and Covariance
This analysis allows us to understand better a number of proposals in the literature aimed at exploiting a background gravitational field to induce baryo/leptogenesis through modified dispersion relations.
in a vierbein frame, where
and where is the projection of the spin connection onto the vierbein basis. Since is the spin current, it appears that in a frame where is constant, this term acts as a chemical potential which in a theory of neutrinos would induce a particle-antiparticle asymmetry. To establish this, it is claimed Singh:2003sp (); Mukhopadhyay:2005gb (); Debnath:2005wk (); Lambiase:2006md (); Lambiase:2011by (); Lambiase:2013haa () that the dispersion relation can be obtained from (55) by considering plane wave solutions of the form444In fact, this solution has no real meaning in general relativity as , or , is not a well defined object except in Minkowski space. In relativity, lives in the tangent plane, but is just an element of the coordinate chart (not a vector) and so it makes no sense to define an “inner product” between the two. giving rise to
for the left and right-handed particles respectively.
However, as we have seen, the true dispersion relations are established in covariant form . In contrast, equation (57) is not covariant, since does not transform as a tensor under the SO(1,3) Lorentz transformations , but rather as
Another way to see is not an SO(1,3) tensor is to note that we can always make it vanish at a point by by choosing an inertial vierbein there. Since tensors are either always zero at a point or never zero there, it cannot be a Lorentz tensor.
We conclude, therefore, that equation (57) is not a valid dispersion relation and has no consequence for gravitational leptogenesis. More generally, any leptogenesis model which requires the non-vanishing of the spin connection (e.g. treating as a chemical potential), depends on working in a non-inertial, accelerating vierbein where . But this is giving information purely on the nature of the acceleration, not revealing the intrinsic covariant physics. While in some situations it is appropriate to consider non-inertial observers, for cosmological applications the appropriate frame in which to measure lepton density is that of an inertial comoving observer. Thus the free Dirac Lagrangian in curved spacetime does not give rise to gravitational leptogenesis.
We should point out, however, that our conclusions apply to Riemannian spacetimes, where the weak equivalence principle holds and the connection vanishes in local inertial frames. Non-Riemannian geometry, spacetimes with torsion, or string backgrounds with additional antisymmetric and dilaton background fields in addition to the metric Ellis:2013gca () may still present interesting generalisations of the picture presented in the last section.
3 Radiatively Induced SEP violation
We now turn to the main topic of this paper, radiatively induced strong equivalence breaking and the realisation of discrete symmetries. Once again, our main focus is on potential applications to gravitational leptogenesis. We are therefore especially interested in the automatic generation by quantum loop effects of operators such as introduced by hand in the model of Davoudiasl et al. Davoudiasl:2004gf () as a C and CP-violating source of matter-antimatter symmetry.
The essential physics behind this effective violation of the strong equivalence principle is readily understood. At the quantum loop level, a particle no longer propagates as a point-like object but is screened by the virtual cloud of particles appearing in its self-energy Feynman diagram Drummond:1979pp (); Ohkuwa:1980jx (); Hollowood:2010bd (); Hollowood:2011yh (). As a result, it acquires an effective size characterised by the Compton wavelength of the virtual particles, causing it to experience gravitational tidal forces through its coupling to the background curvature. Particle propagation at the quantum loop level is therefore described by a mean field whose dynamics are described by the effective action , which gives rise to the quantum-corrected equations of motion
As discussed above, many models of gravitational leptogenesis consider interactions of neutrinos with background curvature Davoudiasl:2004gf (); Lambiase:2006md (); Lambiase:2011by (); Lambiase:2013haa (). It is therefore interesting to perform a thorough investigation of the effective action for neutrinos propagating in a gravitational background. In this section, we examine the effect of quantum loops on the neutrino dispersion relation, and give a careful discussion of C, P and CP symmetries of the 1-loop effective action.
Since we are interested in the propagation of neutrinos, we consider processes of the form shown in figure 3.
The relevant parts of the SM lagrangian, with massless neutrinos, are:
and and are SU(2) and U(1) gauge couplings respectively and , are the and field strengths. Since there are no infrared divergences in the relevant Feynman diagrams and the electron mass contributes only at , it has no qualitative effect on our analysis and may be neglected.
Since we are interested in the free propagation of neutrinos, we need only consider those parts of the effective action quadratic in the mean neutrino field, so that the equations of motion are linear in . We construct the effective action for operators up to third order in derivatives by demanding that it is consistent with general covariance and local Lorentz invariance. This is achieved by constructing all possible neutrino bilinears from the contraction of curvature tensors etc. with gamma matrices and neutrino spinors up to the required number of derivatives. Up to third order in derivatives the following set of operators covers all possible combinations:555With only left-handed neutrinos, there are no dimension 5 operators, since operators of the form or vanish trivially.
One might think that it is possible to construct operators from the Riemann tensor . However the only possible combinations up to third order in derivatives are of the form
and so on, since they must involve only an odd number of gamma matrices, but using the Dirac algebra, and in particular the identities and
it is straighforward to show that these reduce to linear combinations of the operators in (62). Finally using the identity
it can be shown that the final two terms in (62) give a contribution to the action which is a linear combination of other operators:
In summary then, the list of linearly independent bilinears reduces to
and so the most general form of the effective action is
where the can in general be complex. In order to respect the hermiticity of the full electroweak theory, we must demand that the effective action is hermitian and impose
At this point our analysis improves on that of Ohkuwa Ohkuwa:1980jx () who did not impose the requirement of hermiticity in constructing the effective action. The first operator in (69) is hermitian but the remaining operators are not. They have the following hermiticity properties:
We now use these properties and impose the condition (71) to get relations among the . We find that all the coefficients are all real, with the exception of , whose imaginary part must satisfy
Hence with a suitable redefinition of the effective coefficients the most general form of the effective action is
which appear in , therefore form a complete set of linearly independent hermitian operators up to third order in derivatives.666Notice that instead of , and , we could alternatively have used the following basis of independently hermitian operators: , and , but these are less convenient for the subsequent application to the matching conditions and equations of motion.
3.2 Discrete Symmetries
Since the discrete spacetime symmetries P and T single out particular directions in spacetime, we assume the existence of a vector basis with a timelike vector and spacelike vectors which define spatial and temporal directions at each point in the manifold. We can then define P and T transformations locally at each point by:
where the notion is a shorthand defined by
The matrices satisfy and , and , and complex conjugates any complex numbers. This has the consequence that tensor quantities transform as
where the plus and minus sign correspond to P and T respectively. In particular, since we want to transform like under and , it is easy to check that identifying the vector basis above with the vierbein ensures the connection part
(and hence ) transforms like under P and T.
Notice that the arguments of the operators do not transoform as , under P etc. as they do in flat space. The reason is that in flat space, the object is playing the role of a vector (rather than a coordinate) so that should be thought of as an action on the tangent plane, rather than on coordinates. With an understanding of this subtlety, the generalisation to curved space is immediate, and one sees that P and T are only well-defined as actions on vectors in the tangent plane. For example, in the case of a scalar field, the object should be thought of as a vector in the tangent plane at , which transforms according to (85).
We summarise the C, P, CP and CPT properties of the effective operators in a table below. A full derivation of these results can be found in appendix B:
We see that with the exception of (which importantly is CP odd) all the operators respect the CP, T and CPT symmetries of the tree-level EW Lagrangian. It is thus of great interest to investigate the possibility that the CP violating operator is radiatively generated by quantum loop effects. This is particularly pertinent in light of the suggestion by Davoudiasl et al. Davoudiasl:2004gf () that effectively generated C and CP-violating operators such as
give a chemical potential of the form resulting in a gravitationally induced lepton or baryon asymmetry.
We now calculate the coefficients of the curvature terms in the effective action by matching with explicit weak-field Feynman diagram calculations. Since the effective couplings are independent of the choice of geometry, we are free to perform the matching on the most convenient background, providing it is of sufficient generality to distinguish the various terms in the action. The matching is greatly simplified by choosing a conformally flat metric
and conformally rescaled fields
and similarly for the other fields. In terms of the conformally rescaled fields, the Lagrangian becomes