The photon-neutrino interaction in non-commutative gauge field theory and astrophysical bounds

In this letter we propose a possible mechanism of leftand right-handed neutrino couplings to photons, which arises quite naturally in non-commutative gauge field theory. We estimate the predicted additional energy-loss in stars induced by space-time non-commutativity. The usual requirement that any new energy-loss mechanism in globular stellar clusters should not excessively exceed the standard neutrino losses implies a scale of non-commutative gauge theory above the scale of weak interactions. Neutrinos do not carry a U(1) (electromagnetic) charge and hence do not directly couple to Abelian gauge bosons (photons) – at least not in a commutative setting. In the presence of space-time non-commutativity, it is, however, possible to couple neutral particles to gauge bosons via a star commutator. The relevant covariant derivative is D̂μψ̂ = ∂μψ̂ − iκeÂμ ⋆ ψ̂ + iκeψ̂ ⋆ Âμ , (1) with the ⋆–product and a coupling constant κe that corresponds to a multiple (or fraction) κ of the positron charge e. The ⋆–product is associative but, in general, not commutative – otherwise the proposed coupling to the non-commutative photon field Âμ would of course be zero. In (1), one may think of the non-commutative neutrino field ψ̂ as having left charge +κe, right charge −κe and total charge zero. From the perspective of non-Abelian gauge theory, one could also say that the neutrino field is charged in a noncommutative analogue of the adjoint representation with the matrix multiplication replaced by the ⋆–product. From a geometric point of view, photons do not directly couple to the “bare” commutative neutrino fields, but rather modify the non-commutative background. The neutrinos propagate in that background. Kinematically, a decay of photons into neutrinos is, of course, allowed only for off-shell photons. This is still true in a constant or sufficiently slowly varying non-commutative background: Such a background does not lead to a violation of four-momentum conservation, although it may break other Lorentz symmetries. Physically, such a coupling of neutral particles to gauge bosons is possible because the non-commutative background is described by an antisymmetric tensor θ that plays the role of an external field in the theory [1]–[14]. The ⋆–product in (1) is a (non-local) bilinear expression in the fields and their derivatives that takes the form of a series in θ . To lowest order we obtain D̂μψ̂ = ∂μψ̂ + κeθ νρ ∂νÂμ ∂ρψ̂ . A similar expansion (Seiberg-Witten map) exists for the non-commutative fields ψ̂, Âμ in terms of θ μν , ordinary ‘commutative’ fields ψ, Aμ and their derivatives. The scale of non-commutativity ΛNC is fixed by choosing dimensionless matrix elements c = Λ2NCθ μν of order one. Gauge invariance requires that all e’s in the action should be multiplied by κ. To the order considered in this letter, κ can be absorbed in a rescaling of θ, i.e. a rescaling of the definition of ΛNC. The coupling (1) is part of an effective model of particle physics involving neutrinos and photons on non-commutative space-time. It describes the scattering of particles that enter from an asymptotically commutative region into a non-commutative interaction region. The model satisfies the following requirements [1]–[14]: (i) Non-commutative effects are described perturbatively. The action is written in terms of asymptotic commutative fields. (ii) The action is gauge-invariant under U(1)-gauge transformations. (iii) It is possible to extend the model to a non-commutative electroweak model based on the gauge group U(1)×SU(2). An appropriate noncommutative electroweak model with κ = 1 can in fact be constructed with the same tools that were used for the noncommutative standard model of [11]. 1 The action of such an effective model differs from the commutative theory essentially by the presence of star products and Seiberg–Witten (SW) maps. The Seiberg–Witten maps [8] are necessary to express the non-commutative fields ψ̂, Âμ that appear in the action and transform under non-commutative gauge transformations, in terms of their asymptotic commutative counterparts ψ and Aμ. The coupling of matter fields to Abelian gauge bosons is a non-commutative analogue of the usual minimal coupling scheme. The action for a neutral fermion that couples to an Abelian gauge boson in a non-commutative background is S = ∫ dx ( ψ̂ ⋆ iγD̂μψ̂ −mψ̂ ⋆ ψ̂ ) . (2) Here ψ̂(L R ) = ψ(L R ) + eθ νρAρ∂νψ(L R ) and Âμ = Aμ + eθ Aν [ ∂ρAμ − 1 2 ∂μAρ ] is the Abelian NC gauge potential expanded by the Seiberg-Witten map. 2 For a model in which only the neutrino has dual left and right charges, κ = 1 is required by the gauge invariance of the action. Note that instead of Seiberg–Witten map of Dirac fermions ψ one can consider a “chiral” SW map. This SW map is compatible with grand unified models where fermion multiplets are chiral [12].