Single Extra Dimension from $\kappa$-Poincar\'e and Gauge Invariance

We show that $\kappa$-Poincar\'e invariant gauge theories on $\kappa$-Minkowski space with physically acceptable commutative (low energy) limit must be 5-dimensional. General properties of the related actions and possible observable effects are briefly discussed.

Noncommutative (NC) structures are expected to occur at the Planck scale [1] where Quantum Gravity effects become relevant [2]. Among the various noncommutative (quantum) spacetimes, the κ-Minkowski spacetime [3] is believed to be a good candidate to describe the quantum spacetime underlying Quantum Gravity. This noncommutative (quantum) spacetime is known for long to be rigidely linked to the κ-Poincaré algebra [4] coding the quantum version of its relativistic symmetries. This latter already shows up within (2+1)-d gravity with matter as a symmetry of the (effective) Noncommutative Field Theory (NCFT) obtained by integrating out the gravitational degrees of freedom [5] while interesting arguments favoring its role as a symmetry of (3+1)-d quantum gravity (in a particular regime) were given in [6], therefore enforcing the belief that at ultra-high energy, Poincaré invariance as well as Minkowski spacetime should be replaced by their respective κ-deformations.
The phenomenological consequences [2] of these κ-deformations have been examined in many works, dealing e.g. with Doubly Special Relativity [7] or Relative Locality [8]. Since at low energy gauge invariance must supplement Poincaré invariance in any reasonable field theory, one therefore should consider NCFT with both κ-Poincaré invariance and (NC analog of) gauge invariance at energy near the Planck scale. But requiring κ-Poincar invariance endows necessarily the action with a new algebraic property, as recalled below, which depends on the dimension d of the κ-Minkowski space, spoils the cyclicity of the integral involved in the action and prevents the gauge invariance to be achieved, except for a unique value of d, d = 5, stemming from a consistency condition, as we now show.
We use the bicrossproduct basis [4]. Our convention are as in [9]. The d-dimensional κ-Minkowski space M d κ is conveniently described as the algebra of smooth functions on R d with polynomial maximal growth, equipped with the star-product and involution [9,10] (f ⋆g)(x) = dp 0 2π Eq where ǫ : P κ → C is the counit of P d κ , the symbol ⊲ denotes the action of h and φ denotes generically some fields. For instance, using (E ⊲ φ)(x) = φ(x 0 + i κ , x) with E = e −P0/κ , (P µ ⊲ φ)(x) = −i∂ µ φ(x), µ = 0, ..., d − 1 and ǫ(E) = 1, ǫ(P µ ) = 0, one obtains E ⊲ S = S, P µ ⊲ S = 0. It is known that the Lebesgue integral satisfies i.e. d d x is a twisted trace with respect to (1). This trades the usual cyclicity for a KMS property. Indeed, as pointed out in [9,11], the action S defines a KMS weight [12] associated with the (Tomita) group of modular automorphisms [13] whose generator is (4), called the modular twist. For general discussions on physical consequences of KMS property, see [14]. Oneloop properties of κ-Poincaré invariant scalar NC field theories on M 4 κ have been examined in [15], showing soft UV behavior, absence of UV/IR mixing and for some of them vanishing of the beta functions [15]. Had we decided to abandon the κ-Poincaré invariance, then we could have used a cyclic integral w.r.t. the star product, as in e.g. [16]. But, the resulting actions would have had physically unsuitable commutative limits. Note that the loss of cyclicity does not complicate practical calculations: any P d κ -invariant action based on (1)-(4) can be easily represented as a nonlocal field theory involving ordinary integral and commutative product.
We will consider the NC analog of U (1) gauge symmetry [17], [18]. Generalization to larger symmetry follows from a mere adaptation of [18] and would not alter the conclusions of this letter. We look for NC gauge theories on M d κ with polynomial actions depending on the curvature (field strength) of the NC connection (gauge potential), to be characterized below, satisfying two requirements: i) the action is both invariant under P d κ and the NC U (1) gauge symmetry, ii) its commutative limit is physically acceptable (i.e. it coincides with an ordinary field theory). In [19], we have shown that the twisted trace (3) insuring P d κ -invariance restricts the allowed values of d at which such an action may eventually exist. One necessary ingredient is the existence of (at least one) suitable twisted NC differential calculus, the twist being essential. In particular, there is no untwisted differential calculus which can support a gauge invariant action, whatever the dimension of M d κ may be [19], as e.g. the bicovariant differential calculi on κ-Minkowski [20]. The second ingredient related to the NC differential calculus is the construction of a twisted connection and its curvature. Requiring the gauge invariance of the action then amounts to require that the effects of the various twists balance the one of the modular twist (4), resulting in a d-depending consistency relation between all the twists. We now show that gauge invariant actions satisfying i) and ii) can only be obtained from a unique 1-parameter family of twisted derivations of the algebra of the "deformed translations" T κ ⊂ P d κ and only for d = 5, the unique value for which the NC gauge symmetry can be accommodated with the κ-Poincaré invariance.
In the following, there is no summation over the repeated indices in the formulas, unless stated. Consider first a set of d mutually commuting bitwisted derivations Recall that X µ as a bitwisted derivation [19] of M d κ is an element of P d κ satisfying the twisted Leibniz rule: The twists α µ and β µ belong to P d κ and are algebra automorphisms of M d κ (i.e. α µ (a ⋆ b) = α µ (a) ⋆ α µ (b) and the same for β µ ). Hence, to each X µ corresponds a pair of twists (α µ , β µ ). The general framework of NC differential calculi based on such twisted derivations has been characterized in [19]. Here, it will be sufficient to work with the "components" of the 1-form connection and 2-form curvature. Consider now the most general case in which one introduces one twist for each of these components together with related twisted gauge transformations. A general bitwisted connection [25] over the module M ≃ M d κ satisfies [19] where m ∈ M ≃ M d κ and τ µ and ρ µ are automorphisms of M d κ , elements of P d κ . From this follows ∇ µ (a) = A µ τ µ (a) + X µ (a), A µ := ∇ µ (1), (6) where A µ is the NC gauge potential. The most general twisted gauge transformations are [19] where ρ 1,µ and ρ 2,µ are elements of P d κ acting as regular automorphisms [21] for any g in M d κ verifying the unitary relation g † ⋆ g = g ⋆ g † = 1. The group of NC gauge transformations, denoted by U(M d κ ), is therefore the set of unitary elements of M d κ , the NC analog of the U (1) gauge symmetry. Now from algebraic manipulations, one infers that ∇ g µ (a) = A g µ τ µ (a) + X µ (a), ∇ g µ given by (7), defines a connection if the following relations hold true: The general expression of the curvature F µν is obtained from ∇ µ (K µν ∇ ν (a)) − ∇ ν (K νµ ∇ µ (a)) = F µν ⋆ τ µ K µν τ ν (a), where the twist K µν , element of P d κ , acts as an automorphism of M d κ . One finds which is a morphism of (twisted) module if From (10) and (11), a tedious calculation leads to the twisted gauge transformations for F µν given by provided the following relations hold true: τ µ K µν X ν ρ 2,ν (g) = X ν K νµ τ µ ρ 2,µ (g) , We now show that the number of twists is severely restricted, due to compatibility conditions between (α µ , β µ ), the twists of gauge transformations (ρ 1,µ , ρ 2,µ ) and K µν . These conditions insure the stability of the space of connections under gauge transformations and (twisted) gauge covariance of the curvature. Combining (12)-(16) with (18)-(22) yields ρ 1,µ = ρ 1 and ρ 2,µ = ρ 2 , while using the unitary relation, eq. (18) yields ρ 2 = K µν ρ 1 . Hence, K µν = K, so (12) yields β µ = β = K −1 , and (18) yields τ µ = τ . Using ρ 2 = Kρ 1 and differentiating ρ 2 (g † )ρ 2 (g) = 1 by X µ using (5), one can check that (22) is verified. Hence, at this stage only β, α(= τ ) and ρ 2 remain as independent twists. Next, assume first that X µ belongs to T κ . T κ has primitive elements (E, P 0 , P i ) with coproduct ∆(E ⊗E) = E ⊗E, ∆(P 0 ) = P 0 ⊗ I + I ⊗ P 0 , ∆(P i ) = P i ⊗ I + E ⊗ P i . But since τ and β are assumed to be automorphisms of M d κ , their coproduct must be of the Therefore, it follows that τ , β and K must be powers of E, owing to the expression for ∆(E) and thus are regular automorphisms verifying relations similar to (8). Since E commutes with all the elements of P d κ , all the twists β, τ and ρ 2 are mutually commuting.

Now, we look for a gauge invariant action of the form
where in (23) and from now on summation over repeated indices is understood, J µν is an automorphism of M d κ and d d x in S(A) insures that requirement i) is verified. Upon using (17) together with (3) and (4), one easily finds that (23) is invariant under the NC U(M d κ ) gauge transformations provided The combination of eq. (24) with (8) and g ⋆g † = 1 yields J µν = J = E 1−d ρ 2 1 . This, combined with (25), owing to the fact that τ , K, ρ 1,2 commute with each other, gives rise to Using the duality between M d κ and T κ [4] and the above restrictions on the twists, one infers from (5) that the coproduct of any X µ must be of the form ∆(X µ ) = X µ ⊗ τ + β ⊗ X µ . But, as an element of T κ , X µ must be expressible as a finite sum X µ = x mnk E m P n i P k 0 . Then, the combination of these two constraints fixes the allowed twisted derivations in T κ . These are E γ (1 − E), E γ P 0 , E γ P i with respective twists E γ , E γ , E γ+1 where γ is a real parameter. Finally, notice that the use of twisted derivations out of T κ would lead to actions with unusual (physically unsuitable) commutative limits which would not meet requirement ii). To conclude, using (26) and α = τ , one finds that the only physically admissible solution is given by α = E γ , β = E γ+1 . This, plugged into (26), gives finally thus singling out d = 5, independent of γ. This is the unique physical value for the classical dimension at which κ-Poincaré and NC gauge invariance can coexist, selecting in P d κ a unique family of twisted derivations of T κ , given by X This result appears as an interesting physical prediction. It states clearly that κ-Poincaré invariant gauge theories on κ-Minkowski space with physically acceptable commutative limit must be 5-dimensional. As a byproduct, this result gives a rationale based on symmetry constraints for the introduction of an extra (spatial) dimension. Note that any experimental evidence disfavoring the existence of a single extra dimension would render questionable the physical relevance of κ-Poincaré invariant gauge theories and possibly related concepts linked to κ-deformations of Minkowski space-time.
Let us discuss general physical features of κ-Poincaré invariant gauge theories. Consider the coupling of S(A) (23) to a fermion, assuming from now on that A µ is realvalued and ρ 2 = I. The U(M d κ ) gauge invariant action is with / ∇ = γ µ ∇ µ and g 2 1 has mass dimension −1. Gauge invariance of the 2nd term in (28) follows from ψ g = g⋆ψ, ρ 2 = Kρ 1 , K = β −1 combined with (7). The κ → ∞ limit of (28) obviously yields the usual (5-d) QED action, with U(M d κ ) reducing to U (1). By using the formalism of [9], one obtains the kinetic term for A µ : The second term can be gauged away by using the gauge condition X (0) µ A µ = 0. This is done by adding to (28) the gauge-fixing term . The BRST operator s verifies s 2 = 0 and is defined by with C † = −C and sC = b, sb = 0. C, C and b are respectively the ghost, antighost and Stückelberg auxiliary field serving to implement the gauge condition, with respective ghost numbers 1, −1 and 0. Recall that s acts as a derivation w.r.t. the grading defined by the sum of the ghost number and the degree of forms (modulo 2). Upon gauge-fixing, one obtains S kin (A) = 1 , where 0 < γ < 1 insures a suitable decay at large momenta for the propagator. K is naturally identified with the NC analog of the Laplacian. It leads to a deformed energy-momentum relation. For instance, after compactification, the 4-d theory for the zero modes inherits a deformed relation (assuming for simplicity γ = 1 2 and setting E = p 0 ): ( p is a 3-momentum) where κ is still the 5-d (bulk) deformation parameter, hence not necessarily of the order of M P . The gauge-matter interaction can be written as (k a = (k 0 a , k a ), a = 1, 2, 3) showing energy conservation while momentum conservation is "deformed" by the factor e −k 0 1 /κ . The compactification of the extra dimension (e.g. on S 1 or S 1 /Z 2 ) leads to a deformed Kaluza-Klein number conservation law [22] ∼ δ(n 1 + n 2 e −k 0 1 /κ + n 3 ) where the effect of the deformation depends on the magnitude of κ which controls as well possibly observable effects from the NC structure. Similar comments hold for the self-interactions of A µ . Lower bounds from collider experiments [23] for the size of the extra dimension µ −1 within flat extra-dimensional and UED scenarios [22] lead conservatively to µ O(1 − 5)TeV. Embedding the present framework into these scenarios, one expects the well know relation M 2 P ≈ κ 3 1 µ to hold, implying κ O(10 13 )GeV, high enough to suppress typical (κ-depending) NC effects (∼ O( √ s κ )) in collider experiments. More promising are the physical consequences of the deformed dispersion relations which will follow from (28) and (30), resulting in departures from perfect non-dispersiveness. Somewhat similar effects also appear in String Theories or Loop Quantum Gravity [24] stemming from energy-depending velocity for photons (and/or birefringence), resulting in time-delay between 2 photons of different energy (or polarisation) emitted from a distant source. In the present situation, expanding the equation of motion for (28) (with ψ = 0) up to O(κ −2 ) and assuming crudely the kinetic contributions dominate the interaction one yields E 2 − | p| 2 − 1 κ | p| 3 = 0, as the relation in [24] for birefringence. This indicative observation of course deserves to be refined by a careful analysis of peculiar features arising in the κ-expansion of (28), which we have undertaken. This may well push forward the existing bounds on the related mass scales.
J.-C. Wallet thanks F. Lizzi, P. Martinetti, A. Sitarz and A. Wallet for various discussions related to this work.