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4.4 Lorentz violation with gravity in EFT

The previous field theories dealt only with the possible Lorentz violating terms that can be added to the matter sector. Inclusion of gravity into the mix yields a number of new phenomena. Lorentz violating theories with a preferred frame have been studied extensively (cf. [122Jump To The Next Citation Point157Jump To The Next Citation Point277Jump To The Next Citation Point] and references therein), while an extension of the mSME into Riemann-Cartan geometry has been performed in [173Jump To The Next Citation Point]. Ghost condensate models, in which a scalar field acquires a constant time derivative, thereby choosing a preferred frame, were introduced in [34Jump To The Next Citation Point]. Let us first look at the more generic case of [173Jump To The Next Citation Point].

In order to couple Lorentz violating coefficients to fermions, one must work in the vierbein formalism (for a discussion see [272]). In Riemann-Cartan geometry the gravitational degrees of freedom are the vierbein and spin connection which give the Riemann and torsion tensors in spacetime. For the purposes of this review we will set the torsion to zero and work strictly in Riemannian geometry; for the complete Lorentz violating theory with torsion see [173] (for more general reviews of torsion in gravity see [143139]). The low energy action involving only second derivatives in the metric is given by

integral --1--- 4 [ ab abgd ] S = 16pG d xe R - 2/\ + s Rab + t Rabgd , (46)
where e is the determinant of the vierbein, R, Rab, and Rabgd are the Ricci scalar, Ricci tensor, and Riemann tensor, respectively, and /\ is the cosmological constant. G is the gravitational coupling constant, which can be affected by Lorentz violation. Since there is no longer translation invariance, in principle the Lorentz violating coefficients sab and tabgd vary with location, so they also behave as spacetime varying couplings. ab s and abgd t can furthermore be assumed to be trace-free as the trace can be absorbed into G and /\. There are then 19 degrees of freedom left.

The difficulty with this formulation is that it constitutes prior geometry and generically leads to energy-momentum non-conservation, similar to the bimetric model in Section 2.4. Again the matter stress tensor will not be conserved unless very restrictive conditions are placed on sab and tabgd (for example that they are covariantly constant). It is unclear whether or not such restrictions can be consistently imposed in a complicated metric as would describe our universe.

A more flexible approach is to presume that the Lorentz violating coefficients are dynamical, as has been pursued in [122Jump To The Next Citation Point157Jump To The Next Citation Point34185Jump To The Next Citation Point220]. In this scenario, the matter stress tensor is automatically conserved if all the fields are on-shell. The trade-off for this is that the coefficients ab s and abgd t must be promoted to the level of fields. In particularly they can have their own kinetic terms. Not surprisingly, this rapidly leads to a very complicated theory, as not only must sab and tabgd have kinetic terms, but they must also have potentials that force them to be non-zero at low energies. (If such potentials were not present, then the vacuum state of the theory would be Lorentz invariant.) For generic ab s and abgd t, the complete theory is not known, but a simpler theory of a dynamical “aether”, first looked at by [122] and expanded on by [15718510451] has been explored.

The aether models assume that all the Lorentz violation is provided by a vector field ua.12 With this assumption, sab can be written as uaub, and tabgd can always be reduced to an sab term due to the symmetries of the Riemann tensor. The most generic action in D dimensions that is quadratic in fields is therefore

integral ---1-- D V~ --- [ ab m n a ] S = 16pG d x - g R + K mn \~/ au \~/ bu + V(u ua) , (47)
where
ab ab a b a b a b K mn = c1g gmn + c2dmdn + c3dndm + c4u u gmn, (48)
and the sab term has been integrated by parts and replaced with the c,c 1 3 terms. The coefficients c1,2,3,4 are dimensionless constants, R is the Ricci scalar, and the potential a V (u ua) is some function that enforces a non-zero value for a u at low energies. With a proper scaling of coefficients and V this value can be chosen to be unit at low energies. The model of Equation (47View Equation) still allows for numerous possibilities. Besides the obvious choice of which coefficients are actually present, ua can be either spacelike or timelike and in extra dimension scenarios can point in one of the four uncompactified dimensions or the D - 4 compactified ones. At low energies ua acquires an expectation value ua, and there will be excitations dua about this value. Generically, there will be a single massive excitation and three massless ones. It has been argued in [105Jump To The Next Citation Point] that the theory suffers stability problems unless V is of the form c(uau - 1) a, where c is a Lagrange multiplier. The theory is also ghost free with this potential and the further assumption that c1 + c4 < 0 [133Jump To The Next Citation Point]. Assuming these conditions, aether theories possess a set of coupled aether-metric modes which act as new gravitational degrees of freedom that can be searched for with gravitational wave interferometers or by determining energy loss rates from inspiral systems like the binary pulsar. The same scenario generically happens for any tensor field that acquires a VEV dynamically (see Section 7.1), which implies that Lorentz violation can be constrained by the gravitational sector as well as by direct matter couplings.

The aether models use a vector field to describe a preferred frame. Ghost condensate gives a more specific model involving a scalar field. In this scenario the scalar field f has a Lagrangian of the form P (X), where a X = @af @ f. P (X) is a polynomial in X with a minimum at some value X = m, i.e. f acquires a constant velocity at its minimum. In a cosmological setting, Hubble friction drives the field to this minimum, hence there is a global preferred frame determined by the velocity of f. This theory gives rise to the same Lorentz violating effects of aether theories, such as ńĆerenkov radiation and spin dependent forces [33]. In general, systems that give constraints on the coefficients of the aether theory are likely to also yield constraints on the size of the velocity m.


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