Propagation of photons and massive vector mesons between a parity breaking medium and vacuum
aa r X i v : . [ h e p - ph ] O c t Preprint typeset in JHEP style - HYPER VERSION
ICCUB-11-166
Propagation of photons and massive vector mesonsbetween a parity breaking medium and vacuum
A. A. Andrianov ab , S. S. Kolevatov a , R. Soldati c a V.A. Fock Department of Theoretical Physics, Sankt-Petersburg State University,ul. Ulianovskaya, 198504 St. Petersburg, Russia b High Energy Physics Group, Dept. Estructura i Constituents de la Mat`eria andInstitut de Ci`encies del Cosmos, Universitat de Barcelona,Diagonal 647, 08028 Barcelona, Spain c Dipartimento di Fisica, Universit´a di Bologna,Istituto Nazionale di Fisica Nucleare, Sezione di Bologna,via Irnerio 46, 40126 Bologna, ItaliaE-mail: [email protected], [email protected] , [email protected]
Abstract:
The problem of propagation of photons and massive vector mesons in thepresence of Lorenz and CPT invariance violating medium is studied when the parity-oddmedium is bounded by a hyperplane separating it from the vacuum. The solutions in bothhalf-spaces are carefully discussed and in the case of space-like boundary stitched on theboundary with help of the Bogolubov transformations provided by the space-like Chern-Simons vector. The presence of two different Fock vacua is shown and the probabilityamplitude for transmission of particles from vacuum to parity breaking medium is calcu-lated. We have also found classical solutions and showed that the results are consistentwith ones obtained by canonical quantization formalism. In the cases, both of entranceto and of escaping from parity-odd medium, the probabilities for reflecting and passingthrough were found for each polarization using the classical solutions. Finally, the prop-agator for each polarization is obtained in the momentum space. Boundary effects underconsideration are of certain importance for registration of local parity violation in the finitevolume of heavy ion fireball and/or of a star with cold axion condensate.
Keywords:
Local parity and Lorentz symmetry breaking, photon decay, Space-TimeSymmetries. ontents
1. Introduction: possible physics of time- or space-dependent pseudoscalarcondensate 12. Vector Fields in a Pseudoscalar Background 3
3. The Bogolyubov Transformations 104. CS Electrodynamics for a space-like CS vector 14
5. Conclusions and outlook 21
1. Introduction: possible physics of time- or space-dependent pseudoscalarcondensate
The limits of validity of fundamental laws in macroscopic space-time Physics have beenattracting more interest following remarkable experimental improvements, both in labora-tory research and in astrophysics [1]–[10]. Specifically in Quantum Electrodynamics theinterest to possible Lorentz and CPT Invariance Violation (LIV for short) was raised upafter the seminal paper [11] where the very possibility to have a constant vector backgroundgenerating Lorentz and CPT parity breaking in the large scale universe was conjecturedand falsified. The latter was employed to modify QED supplementing it with the Chern-Simons (CS) parity-odd lagrangian spanned on a constant CS vector. Different theoreticalways for derivation of the Carroll-Field-Jackiw Electrodynamics from the fermion matterinteraction to a constant axial vector background (condensate of axial vector field, axioncondensate or a gravity torsion) were considered and discussed in [12]–[24]. Later the var-ious aspects of its signatures were discussed [25]– [34] although this sort of LIV has notbeen yet detected [35]–[37]. On the other hand, spontaneous Lorentz symmetry breakingmay occur after condensation of massless axion-like fields [25],[35]–[43] at large space scalescomparable with star and galaxies sizes.Thus the failure in detecting a tiny violation of Lorentz invariance and parity in thelarge scale universe does not exclude such effects at the level of galaxies and stars. For– 1 –nstance, cold relic axions resulting from vacuum misalignment[44, 45] in the early universeis a viable candidate to dark matter. If we assume that cold axions are the only contributorsto the matter density of the universe apart from ordinary baryonic matter its density mustbe [46, 47, 48] ρ ≃ − gcm − ≃ − GeV . Of course dark matter is not uniformlydistributed, its distribution presumably follows that of visible matter. One may thinkalso of an axion background accumulated by very dense stars like neutron ones or evenof bosonic axion stars [49]. Slowly varying axion background or condensed axions can bein principle discovered as such a background induces the high-energy photon decays intodilepton pairs [50] and, in turn, photon emission by charged particles [51, 52, 53, 54].Another interesting area for observation of parity breaking is the heavy ion physics.Recently several experiments in heavy ion collisions have indicated an abnormal yield oflepton pairs of invariant mass < e + e − and dimuon pairs and a possible explanation of this enhancement is outlinedin [62]. It was conjectured that the effect may be a manifestation of local parity breaking(LPB) in colliding nuclei due to generation of pseudoscalar, isosinglet or neutral isotriplet,classical background whose magnitude depends on the dynamics of the collision. Theo-retical reasons to generate an isotriplet pseudoscalar condensate at large baryon densitieshave been given in [63, 64]. In addition to, in [65, 66, 67, 68] it has been suggested thatfor peripheral interactions a complementary phenomenon of the so-called Chiral MagneticEffect should occur. It is triggered by an isosinglet pseudoscalar background as the re-sult of large-scale fluctuation of topological charge and has been studied by lattice QCDsimulations [69] and nearly detected in the STAR experiments on RHIC [70, 71].In the occurrences of axion-like background in astrophysics or heavy ion physics theexistence of a boundary between the parity-odd medium and the vacuum is quite essential.For star condensed axions there is evidently a boundary where axion background disappearsand photons distorted by it escape to vacuum. In the presence of such a space boundary notall the photons penetrate it and partially a reflection arises which will be described in detailsin Sec.4.3. As well in heavy ion collisions the outcome of the photon/vector meson decays,say, into lepton pairs generated by slowly decreasing pseudoscalar background inside of thefireball can be normally registered in vacuum or after freeze-out when dilepton pairs areoutside of the parity breaking medium.Thus the examination of how an axion/pion background in a bounded volume caninfluence on photon and massive vector mesons propagating through a boundary to vacuumrepresents a real interest for detecting parity odd properties of a medium. In this workwe analyze a simplified model with two semi-infinite spaces, one of which is filled by apseudoscalar matter and another one to be a normal vacuum. Such a model may shedlight on the processes of photon and vector meson emission in the crust of axion reachneutron stars or on the boundary of fireballs when wave lengths of vector states are muchless than the size of a distorted vacuum with parity breaking background. It is defined inthe next section and its spectrum depending on polarizations and canonical quantizationis examined in details. In Sec.3 we show that for a purely space-arrowed CS vector such– 2 –n electrodynamics can be consistently quantized having the two different vacuum statesrelated by Bogolyubov transformation. In Sec.4 the matching of vector boson fields betweenparity-odd CS medium and vacuum is elaborated in details for a space-like boundary. Thetransmission and reflection of massive vector mesons when propagating outside of parity-odd medium to vacuum and in the opposite direction are calculated. In Subsec. 4.5the Green function for this propagation is reconstructed. Conclusions and perspectives forobservation of parity odd media are summarized in the last section where also the geometryof distorted photon decays is discussed for how to serve as a possible signature for detectingparity breaking media by the anisotropy in emission of dilepton pairs.
2. Vector Fields in a Pseudoscalar Background
We start from the Lagrange density which describes the propagation of a vector field inthe presence of a pseudoscalar axion-like background, L = − F αβ ( x ) F αβ ( x ) − g F µν ( x ) e F µν ( x ) a cℓ ( x ) /M + m A ν ( x ) A ν ( x ) + A µ ( x ) ∂ µ B ( x ) + κ B ( x ) , (2.1)where A µ and a cℓ stand for the vector and background pseudoscalar fields respectively, e F µν = ε µνρσ F ρσ is the dual field strength, while B is the auxiliary St¨uckelberg scalarfield with κ ∈ R . The positive dimensionless coupling g > M ≫ m do specify the intensity and the scale of the pseudoscalar-vector interaction.Notice that we have included the Proca mass term for the vector field because, as it isdiscussed in [62], the latter is required to account for the strong interaction effects inheavy ion collisions supported by massive vector mesons ( ρ, ω, . . . ) in addition to photons.Moreover, as thoroughly debated in [72], the mass term for the vector field appears tobe generally necessary to render the dynamics self-consistent in the presence of a Chern-Simons lagrangian and is generally induced by radiative corrections from the fermionicmatter lagrangian . The auxiliary St¨uckelberg lagrangian, which further violates gaugeinvariance beyond the mass term for the vector field, has been introduced to provide – justowing to the renowned St¨uckelberg trick – the simultaneous occurrences of power countingrenormalizability and perturbative unitarity for a general interacting theory. Moreover, itspresence allows for a smooth massless limit of the quantized vector field. We shall consider a slowly varying classical pseudoscalar background of the kind a cℓ ( x ) = Mg ζ λ x λ θ ( − ζ · x ) (2.2)where θ ( · ) is the Heaviside step distribution, in which a fixed constant four vector ζ µ withdimension of a mass has been introduced, in a way to violate Lorentz and CPT invariancein the Minkowski half space ζ · x < . In what follows we shall suppose that ζ = 0 . If we We also leave the room for the photon mass generation in a plasma like medium. – 3 –ow insert the specific form (2.2) of the pseudoscalar background in the pseudoscalar-vectorcoupling lagrangian we can write − F µν ( x ) e F µν ( x ) ζ λ x λ θ ( − ζ · x ) = ζ µ A ν ( x ) e F µν ( x ) θ ( − ζ · x ) − ∂ µ h A ν ( x ) e F µν ( x ) ζ λ x λ θ ( − ζ · x ) i (2.3)The very last term in the RHS of the above equality is evidently a boundary term, itscontribution to the Action being reduced for the Gauß theorem to Z Ω d x ∂ µ h A ν ( x ) e F µν ( x ) ζ λ x λ θ ( − ζ · x ) i = Z ∂ Ω d σ µ A ν ( x ) e F µν ( x ) ζ λ x λ θ ( − ζ · x )where Ω is an arbitrary domain of the Minkowski space-time that is bounded by the initial and final three dimensional space-like oriented surfaces ∂ Ω = Σ ı ∪ Σ f . Hence the boundaryterm won’t contribute to the Euler-Lagrange field equations iff A ν ( x ) e F µν ( x ) ζ λ x λ θ ( − ζ · x ) (cid:12)(cid:12)(cid:12) Σ ı = A ν ( x ) e F µν ( x ) ζ λ x λ θ ( − ζ · x ) (cid:12)(cid:12)(cid:12) Σ f ≡ ζ · x < . In such a circumstance we can derivethe field equations from the equivalent Lagrange density L = − F αβ ( x ) F αβ ( x ) + ζ µ A ν ( x ) e F µν ( x ) θ ( − ζ · x )+ m A ν ( x ) A ν ( x ) + A µ ( x ) ∂ µ B ( x ) + κ B ( x ) (2.4)in which the gauge invariance is badly broken by all the terms but the one, i.e. theMaxwell’s radiation lagrangian. Then the field equations read ∂ λ F λν + m A ν + ζ α e F αν + ∂ ν B = 0 for ζ · x < ∂ λ F λν + m A ν + ∂ ν B = 0 for ζ · x > ∂ ν A ν = κ B (2.5)After contraction of the first pair of the above set of field equations with ∂ ν we find (cid:0) (cid:3) + κ m (cid:1) B ( x ) = 0 (2.6)whence it follows that the auxiliary St¨uckelberg field is always a decoupled unphysical realscalar field, which is never affected by the pseudoscalar classical background ∀ κ ∈ R . Fromnow on we shall select the simplest choice κ = 1 that leads to the Klein-Gordon equationfor the auxiliary field, together with (cid:3) A ν ( x ) + m A ν ( x ) = ε ναρσ ζ α ∂ ρ A σ ( x ) for ζ · x < (cid:3) A ν ( x ) + m A ν ( x ) = 0 for ζ · x > ∂ ν A ν ( x ) = B ( x ) (cid:0) (cid:3) + m (cid:1) B ( x ) = 0 (2.7)In order to find the most general solution of the above linear equations (2.7) we extendthe equations in half-spaces to the entire plane and turn to the momentum space A ν ( x ) = Z d k (2 π ) / a ν ( k ) e − ik · x B ( x ) = Z d k (2 π ) / b( k ) e − ik · x – 4 –o, the second and third line equations (2.7) for entire space-time ( (cid:3) A ν ( x ) + m A ν ( x ) = 0 ∂ ν A ν ( x ) = B ( x ) (cid:0) (cid:3) + m (cid:1) B ( x ) = 0 (2.8)can be written in the momentum space, (cid:2) g λν (cid:0) k − m (cid:1) − k λ k ν (cid:3) a λ ( k ) + i k ν b( k ) = 0 k λ a λ ( k ) = i b( k ) (cid:0) k − m (cid:1) b( k ) = 0 (2.9)The general solutions of these field equations are the well known Proca-St¨uckelberg vectorand auxiliary ghost scalar quantum free fields, viz., A µ ( x ) = A µ PS ( x ) − ∂ µ B ( x ) /m (2.10) A µ PS ( x ) = Z d k X r =1 h a k , r u µ k , r ( x ) + a † k , r u µ ∗ k , r ( x ) i ∂ µ A µ PS ( x ) = 0 (2.11) B ( x ) = m Z d k h b k u k ( x ) + b † k u ∗ k ( x ) i (2.12) u ν k , r ( x ) = [ (2 π ) ω k ] − / e νr ( k ) exp {− i ω k x + i k · x } ( r = 1 , , u k ( x ) = [ (2 π ) ω k ] − / exp {− i ω k x + i k · x } ω k ≡ p k + m (2.14)where the creation destruction operators fulfill the canonical commutation relations[ a k , r , a † k ′ , s ] = δ ( k − k ′ ) δ rs [ b † k , b k ′ ] = δ ( k − k ′ ) (2.15)all the remaining commutators being equal to zero. The three linear polarization realvectors do satisfy the orthonormality and closure relations on the mass shell k = m :namely, k µ e µr ( k ) = 0 − g µν e µr ( k ) e νs ( k ) = δ rs X r =1 e µr ( k ) e νr ( k ) = − g µν + k µ k ν m (2.16)while the vector plane wave functions fulfill ∂ µ u µ k , r ( x ) = 0 ∀ k ∈ R , r = 1 , , (cid:16) u µ k , r , u ν p , s (cid:17) = Z d x u µ ∗ k , r ( t, x ) i ←→ ∂ u ν p , s ( t, x ) = δ ( k − p ) e µr ( k ) e νs ( p ) = − (cid:16) u µ ∗ k , r , u ν ∗ p , s (cid:17) (2.18) (cid:16) u µ ∗ k , r , u ν p , s (cid:17) = (cid:16) u µ k , r , u ν ∗ p , s (cid:17) = 0 (2.19)For the half-space ζ · x < ( (cid:2) g λν (cid:0) k − m (cid:1) − k λ k ν + i ε λναβ ζ α k β (cid:3) a λ ( k ) + i k ν b( k ) = 0 k λ a λ ( k ) = i b( k ) (cid:0) k − m (cid:1) b( k ) = 0 (2.20)– 5 –he general solutions of the equations for the Maxwell-Chern-Simons free quantumfield, have been extensively discussed and applied in [72, 73] for the massive case and in[14] for the massless case. However, in the light of the present applications it’s better toshortly overview this topic. To be definite, let us first recall the construction of the so called chiral or birefringentpolarization vectors for the Maxwell-Chern-Simons (MCS) vector field. Here we aim todevelop a rather general frame which could allow to readily interplay among the massive,massless, temporal and spatial cases, as we shall specify in the sequel. The starting pointis the rank-two symmetric matrix [72] S νλ ≡ ε µναβ ζ α k β ε µλρσ ζ ρ k σ = δ νλ D + k ν k λ ζ + ζ ν ζ λ k − ζ · k ( ζ λ k ν + ζ ν k λ ) (2.21)where D ≡ ( ζ · k ) − ζ k = S νν in such a manner that we find S νλ ζ λ = S νλ k λ = 0 S µν S νλ = D S µλ S νν = 2 D (2.22)while S µλ ε λναβ ζ α k β = D ε µναβ ζ α k β (2.23)Notice that for a temporal Chern-Simons vector ζ µ = ( ζ , , ,
0) we find D = ζ k ≥ , while for a spatial one like e.g. ζ µ = (0 , − ζ x , ,
0) we get D = ζ x ( k − k − k ) ≥ k ≥ k + k which lies inside the causal cone k ≥
0. Then, toour purpose, it is convenient to introduce the two orthonormal, one dimensional, hermitianprojectors π µν ± ≡ S µν D ± i ε µναβ ζ α k β D − = (cid:0) π νµ ± (cid:1) ∗ = (cid:0) π µν ∓ (cid:1) ∗ ( D >
0) (2.24)It is worthwhile to observe that the addendum involving the Levi-Civita symbol is alwaysimaginary for a temporal Chern-Simons vector ζ µ = ( ζ , , , ζ µ = (0 , − ζ x , ,
0) it is imaginary in the domain k ≥ k + k . The above pair of chiralprojectors actually encodes the occurrence of birefringence or vacuum Faraday’s effect andenjoys the following useful properties ∀ k µ = ( k , k ): namely, π µν ± ζ ν = π µν ± k ν = 0 g µν π µν ± = 1 (2.25) π µλ ± π ± λν = π µ ± ν π µλ ± π ∓ λν = 0 (2.26) π µν + + π µν − = S µν / D π µν + − π µν − = iε µναβ ζ α k β D − Actually it corresponds to the light cone section D ≡ { k k } ∩ { k = 0 } . – 6 – couple of chiral polarization vectors for the Maxwell-Chern-Simons free vector field canbe constructed out of some tetrad of constant quantities ǫ ν , taking into account that wehave π µλ ± ǫ µ ǫ λ = D ǫ + ζ ( ǫ · k ) = [ ( ζ · k ) − ζ k ] ǫ + ζ ( ǫ · k ) (2.27)For example, if we choose ǫ ν = (0 , , , ζ µ =(0 , − ζ x , ,
0) we find π µλ ± ǫ µ ǫ λ = k − k k − k − k = − k k + k − k < ∀ k µ = ( k , k ) , k ≥ k − k . Alternatively, had we opted for the symmetric choice ¯ ǫ ν = (0 , , , / √
3, then for thetemporal Chern-Simons vector ζ µ = ( ζ , , ,
0) we get π µλ ± ¯ ǫ µ ¯ ǫ λ = −
12 + ( k + k + k ) k < ∀ k µ = ( k , k )Hence in both cases we can always build up a pair of space-like, complex, birefringent,chiral polarization vectors ε µ ± ( k ) = ( π µλ ± ǫ λ (cid:2) ( k − k ) / ( k + k − k ) (cid:3) − for ζ µ = (0 , − ζ x , , π µλ ± ¯ ǫ λ (cid:2) − ( k + k + k ) / k (cid:3) − for ζ µ = ( ζ , , ,
0) (2.28)For D > ε µ ∗± ( k ) = ε µ ∓ ( k ) − g µν ε µ ∗± ( k ) ε ν ± ( k ) = 1 g µν ε µ ∗± ( k ) ε ν ∓ ( k ) = 0as well as the closure relations ε µ ∗ + ( k ) ε ν + ( k ) + ε µ ∗− ( k ) ε ν − ( k ) = ε µ − ( k ) ε ν + ( k ) + ε µ + ( k ) ε ν − ( k ) = D − S µν (2.29)In order to obtain the normal modes expansion of the MCS quantum field, let’s intro-duce the kinetic 4 × K with elements K λν ≡ g λν (cid:0) k − m (cid:1) + iε λναβ ζ α k β (2.30)which satisfies K λν = K ∗ νλ Now we are ready to find the general solution of the free field equations (2.20) . As amatter of fact, from the relationships (2.25) and (2.26) we readily obtain K µν ε ν ± ( k ) = h δ µν (cid:0) k − m (cid:1) + √ D (cid:0) π µ + ν − π µ − ν (cid:1) i ε µ ± ( k )= (cid:16) k − m ± √ D (cid:17) ε µ ± ( k ) (2.31)– 7 –hich shows that the polarization vectors of positive and negative chiralities respectivelyare solutions of the vector field equations for ζ · x < k µ ± = ( ω k , ± , k ) ε µ ± ( k , ζ ) = ε µ ± ( k ± ) (cid:0) k ± = ω k , ± (cid:1) (2.32) ω k , ± = p k + m ± ζ | k | for ζ µ = ( ζ , , , r k + m + ζ x ± ζ x q k + m + ζ x for ζ µ = (0 , − ζ x , ,
0) (2.33)It is worthwhile to enlighten on the above relationships with some comments and remarks.Firstly, the four momentum k µ − = ( ω k , − , k ), that specifies the polarization vector ε µ − ( k − )of negative chirality, turns out to stay within the causal cone k − > | k | < m / | ζ | in the temporal case or | k | < m / | ζ x | for the spatial Chern-Simons vector [14, 72, 74]. Furthermore, the above birefringent dis-persion relations fulfilled by the chiral polarizations do admit quite different massless limitsas discussed in [14]. As a matter of fact, in the spatial case ζ µ = (0 , − ζ x , ,
0) the masslesslimit is smooth and safe since both angular frequencies ω k , ± keep real and the correspond-ing group velocities ∇ k ω k , ± have modulus less than one in natural units. Conversely, forthe temporal case ζ µ = ( ζ , , ,
0) the massless limit is troublesome because the angularfrequency ω k , − is imaginary in the infrared and both the related group velocities havemodulus which is always larger than one. Last but not least, it is crucial to realize that thechiral polarizations of the Maxwell-Chern-Simons quanta have nothing to share with theelliptic polarizations of the electromagnetic radiation fields since, for instance, the formerundergo vacuum birefringence, while the latter never do it.To complete our construction of a basis we suitably introduce the further pair oforthonormal polarization vectors, respectively the so called scalar and longitudinal polar-ization real vectors ε µS ( k ) ≡ k µ √ k ( k > ε µL ( k ) ≡ (cid:0) D k (cid:1) − (cid:0) k ζ µ − k µ ζ · k (cid:1) ( k > ∨ D > k µ ε µL ( k ) = 0 k µ ε µS ( k ) = √ k ( k > g µν ε µS ( k ) ε νS ( k ) = 1 g µν ε µL ( k ) ε νL ( k ) = − g µν ε µS ( k ) ε νL ( k ) = g µν ε µS ( k ) ε ν ± ( k ) = g µν ε µL ( k ) ε ν ± ( k ) = 0 (2.38)Hence we have at our disposal ∀ k µ with k > ∨ D > ε µA ( k ) = k µ / √ k for A = S (cid:0) k ζ µ − k µ ζ · k (cid:1) / √ D k for A = Lε µ ± ( k ± ) for A = ± ( k > ∨ D > × g AB = g AB ≡ − − − ( A, B = S, L, + , − ) (2.40)then we can write the full orthogonality and closure relations g µν ε µ ∗ A ( k ) ε νB ( k ) = g AB g AB ε µ ∗ A ( k ) ε νB ( k ) = g µν (2.41)together with the transversality relation k ν ε νA ( k ) = √ k δ AS (2.42)It is very important to keep in mind [72, 73] that the on mass shell polarization vector ε µ − ( k − ) with k − > k keeps below the momentumcutoff, i.e., inside the large momentum sphere | k | < m /ζ ≡ Λ, where ζ stands for either | ζ | or | ζ x | .Now, in order to fully implement the canonical quantization of the MCS massive vectorfield for the especially simple choice κ = 1, it is convenient to introduce the polarized planewaves according to v ν k A ( x ) = (cid:2) (2 π ) ω k A (cid:3) − ε νA ( k ) exp {− iω k A x + i k · x } (2.43)where the dispersion relation for the scalar and longitudinal frequencies is the covariantone, viz., ω k S = ω k L = p k + m ≡ ω k so that we can write k ν ε νS ( k ) = m i∂ ν v ν k S ( x ) = u k ( x ) (2.44)It follows therefrom that the general solution of the Euler-Lagrange equations (2.7) for thequantized massive vector field when κ = 1 and ζ · x < A ν ( x ) = A ν CS ( x ) − ∂ ν B ( x ) /m (2.45) A ν CS ( x ) = Z d k X A = ± ,L h c k ,A v ν k A ( x ) + c † k ,A v ν ∗ k A ( x ) i (2.46) B ( x ) = m Z d k h b k u k ( x ) + b † k u ∗ k ( x ) i (2.47)where the canonical commutation relations holds true, viz., h c k ,A , c † k ′ ,A ′ i = − g AA ′ δ ( k − k ′ ) c k ,S = b k (2.48)all the other commutators being equal to zero. According to equations (2.7) and (2.20) weobtain B ( x ) = − i Z d k k ν h c k S v ν k S ( x ) − c † k S v ν ∗ k S ( x ) i k = ω k == m Z d k h b k u k ( x ) + b † k u ∗ k ( x ) i k = ω k (2.49)– 9 –n such a manner that the physical Hilbert space H phys with positive semi-definite metric,for the MCS massive quanta, is selected out from the Fock space F by means of thesubsidiary condition B ( − ) ( x ) | phys i = 0 ∀ | phys i ∈ H phys ⊂ F (2.50)On the other side, it turns out that the physical MCS massive quanta are created out ofthe Fock vacuum by the creation part of the quantized physical massive MCS vector field A ν CS ( x ), with the standard nonvanishing canonical commutation relations h c k ,A , c † k ′ ,A ′ i = δ AA ′ δ ( k − k ′ ) A, A ′ = L, ± all the other commutators being equal to zero. Notice that the MCS massive 1-particlestates of definite spatial momentum k do exhibit three polarization states, i.e., one linearlongitudinal polarization of real vector ε νL ( k ) with dispersion relation k = m and twochiral transverse states with complex vectors ε ν ± ( k ± ) and dispersion relations (2.33), thenegative chirality states ε ν − ( k − ) being well defined only for | k | < Λ ⇔ k − >
3. The Bogolyubov Transformations
The Proca-St¨uckelberg vector field and the Maxwell-Chern-Simons massive vector field faceone another at the hyperplane ζ · x = 0. Hence locality of the quantized wave fields doesrequire equality on the surface separating the classical pseudoscalar background from thevacuum: namely, δ ( ζ · x ) (cid:2) A µ PS ( x ) − A µ CS ( x ) (cid:3) = 0 , (3.1)while the auxiliary unphysical field B ( x ) is not at all affected by the presence of theboundary ζ · x = 0. Let us first discuss the case of a spatial Chern-Simons vector ζ µ =(0 , − ζ x , ,
0) so that δ ( ζ · x ) = ζ − x δ ( x ) and set such objects: ˆ k = ( ω, k , k ), ˆ x = ( x , x , x ): ˆ k · ˆ x = − ωx + k x + k x . We can write solution in form: A µ PS ( x ) = Z d ˆ k θ ( ω − k ⊥ − m ) X r =1 h a ˆ k , r u µ ˆ k , r ( x ) + a † ˆ k , r u µ ∗ ˆ k , r ( x ) i , (3.2) ∂ µ A µ PS ( x ) = 0 u ν ˆ k , r ( x ) = [ (2 π ) k ] − / e νr (ˆ k ) exp { i k x + i ˆ k · ˆ x } ( r = 1 , , k = q ω − k ⊥ − m (3.3)for x >
0. where the creation-destruction operators fulfill the canonical commutationrelations [ a ˆ k , r , a † ˆ k ′ , s ] = δ (ˆ k − ˆ k ′ ) δ rs (3.4)all the remaining commutators being equal to zero. The three linear polarization realvectors do satisfy the orthogonality and closure relations on the mass shell k = m :– 10 –amely, k µ e µr (ˆ k ) = 0 − g µν e µr (ˆ k ) e νs (ˆ k ) = δ rs X r =1 e µr (ˆ k ) e νr (ˆ k ) = − g µν + k µ k ν m (3.5)And for x < A ν CS ( x ) = Z d ˆ k θ ( ω − k ⊥ − m ) X A h c ˆ k,A v ν ˆ k A ( x ) + c † ˆ k,A v ν ∗ ˆ k A ( x ) i (3.6) A ∈ { L, + , −} v ν ˆ k A ( x ) = (cid:2) (2 π ) k A (cid:3) − ε νA ( k ) exp { ik x + i ˆ k · ˆ x } (3.7)where the canonical commutation relations holds true, viz., h c ˆ k,A , c † ˆ k ′ ,A ′ i = − g AA ′ δ (ˆ k − ˆ k ′ ) (3.8)all the other commutators being equal to zero.So, boundary conditions become Z d ˆ k θ ( ω − k ⊥ − m ) ×× (X A h c ˆ k,A v µ ˆ k,A (ˆ x ) + c † ˆ k,A v µ ∗ ˆ k,A (ˆ x ) i − X r =1 h a ˆ k , r u µ ˆ k,r (ˆ x ) + a † ˆ k , r u µ ∗ ˆ k,r (ˆ x ) i) = 0 (3.9)Now, we suggest v ν ˆ k,A (ˆ x ) = X s =1 h α sA (ˆ k ) u ν ˆ k,s (ˆ x ) − β sA (ˆ k ) u ν ∗ ˆ k,s (ˆ x ) i (3.10)On the one hand we can find D u µ ˆ p,r | v ν ˆ k,A E ≡ − i Z d ˆ x u µ ∗ ˆ p,r ( t, y, z ) ←→ ∂ v ν ˆ k,A ( t, y, z )= − k A + k √ k A k exp { ix ( k − k A ) } δ (ˆ k − ˆ p ) e µr (ˆ k ) ε νA (ˆ k ) (3.11)On the other hand we obtain D u µ ˆ p,r | u λ ˆ k,s E = δ (ˆ k − ˆ p ) e µr (ˆ k ) e λs (ˆ k ) (3.12) D u µ ˆ p,r | u λ ∗ ˆ k,s E = 0 (3.13)and thereby D u µ ˆ p,r | v ν ˆ k,A E = δ (ˆ k − ˆ p ) X s =1 α sA (ˆ k ) e νs (ˆ k ) e µr (ˆ k ) (3.14) D u µ ∗ ˆ p,r | v ν ˆ k,A E = 0 (3.15)– 11 – comparison yields − k A + k √ k A k exp { ix ( k − k A ) } ε νA (ˆ k ) = X s =1 α sA ( t, ˆ k ) e νs (ˆ k ) (3.16)the solution of which is provided by α sA (ˆ k ) = − g µν ε µA (ˆ k ) e νs (ˆ k ) k A + k √ k A k exp { ix ( k − k A ) } (3.17)But for our boundary x = 0, so exp { ix ( k − k A ) } = 1 and we can write α sA (ˆ k ) = − g µν ε µA (ˆ k ) e νs (ˆ k ) k A + k √ k A k (3.18)as it can be readily checked by direct substitution.Let’s evaluate the matrix elements X s =1 h α sA (ˆ k ) α ∗ sB (ˆ k ) − β sA (ˆ k ) β ∗ sB (ˆ k ) i A, B = L, ± First we find g µν ε µA (ˆ k ) g ικ ε ι ∗ B (ˆ k ) X s =1 e νs (ˆ k ) e κs (ˆ k )= ε µA (ˆ k ) ε ι ∗ B (ˆ k ) (cid:18) − g ιµ + k ι k µ m (cid:19) = − g ιµ ε µA (ˆ k ) ε ι ∗ B (ˆ k ) = δ AB where use has been made of the transversality condition (2.42), as well as the orthonor-mality relations (2.19), together with the fact that the covariant linear polarization vectorshave been chosen to be real. Hence we eventually obtain X s =1 α sA (ˆ k ) α ∗ sB (ˆ k ) = δ AB ( k A + k ) k A k X s =1 β sA (ˆ k ) β ∗ sB (ˆ k ) = δ AB ( k A − k ) k A k Subtraction of the above expressions yields the customary relation X s =1 h α sA (ˆ k ) α ∗ sB (ˆ k ) − β sA (ˆ k ) β ∗ sB (ˆ k ) i = δ AB A, B = L, ± (3.19)and by making quite analogous manipulations one can readily check that the further usualBogolyubov relations X s =1 h α sA (ˆ k ) β ∗ sB (ˆ k ) − β sA (ˆ k ) α ∗ sB (ˆ k ) i = 0 A, B = L, ± (3.20)– 12 – X s =1 h α sA (ˆ k ) β sB (ˆ k ) − β sB (ˆ k ) α sA (ˆ k ) i = 0 A, B = L, ± (3.21)Turning back to the boundary condition (3.9) and taking the Bogolyubov transforma-tion (3.10) into account, we can write the operator equalities a ˆ k, r = X A = ± ,L h α rA (ˆ k ) c ˆ k,A − β ∗ rA (ˆ k ) c † ˆ k,A i (3.22) c ˆ k,A = X r =1 h α ∗ Ar (ˆ k ) a ˆ k, r + β ∗ Ar (ˆ k ) a † ˆ k, r i (3.23)From the canonical commutation relations we obtain h a ˆ k, r , a † ˆ p, s i = δ (ˆ k − ˆ p ) δ rs = X A,B = ± ,L h α rA (ˆ k ) c ˆ k,A − β ∗ rA (ˆ k ) c † ˆ k,A , α ∗ sB (ˆ p ) c † ˆ p,B − β sB (ˆ p ) c ˆ p,B i = X A = ± ,L h α rA (ˆ k ) α ∗ sA (ˆ p ) − β rA (ˆ k ) β ∗ sA (ˆ p ) i δ (ˆ k − ˆ p )and consequently X A = ± ,L h α rA (ˆ k ) α ∗ sA (ˆ k ) − β rA (ˆ k ) β ∗ sA (ˆ k ) i = δ rs (3.24)The null commutators h a ˆ k, r , a ˆ p, s i = h a † ˆ k, r , a † ˆ p, s i = 0 lead to the further relations X A = ± ,L h α rA (ˆ k ) β ∗ sA (ˆ k ) − β rA (ˆ k ) α ∗ sA (ˆ k ) i = 0 (3.25) X A = ± ,L h α rA (ˆ k ) β sA (ˆ k ) − β rA (ˆ k ) α sA (ˆ k ) i = 0 (3.26)There are two different Fock vacua: namely, a ˆ k, r | i = 0 c ˆ k,A | Ω i = 0whence c ˆ k,A | i = X r =1 β ∗ Ar (ˆ k ) a † ˆ k, r | i and consequently h | c † ˆ p,B c ˆ k,A | i = X r,s =1 h | a ˆ p,s a † ˆ k, r | i β Bs (ˆ p ) β ∗ Ar (ˆ k )= δ (ˆ k − ˆ p ) X r =1 β Br (ˆ k ) β ∗ Ar (ˆ k ) = δ (ˆ k − ˆ p ) δ AB ( k A − k ) k A k – 13 –n turn we evidently obtain a ˆ k, r | Ω i = − X A = L, ± β ∗ rA (ˆ k ) c † ˆ k,A | Ω i (3.27)that yields h Ω | a † ˆ p,s a ˆ k,r | Ω i = X A,B = L, ± h Ω | c ˆ p,A c † ˆ k, B | Ω i β Bs (ˆ p ) β ∗ Ar (ˆ k )= δ (ˆ k − ˆ p ) X A = L, ± β As (ˆ k ) β ∗ Ar (ˆ k )Moreover we get h | a ˆ p,s c ˆ k,A | i = h | a ˆ p,s X r =1 h α ∗ Ar (ˆ k ) a ˆ k, r + β ∗ Ar (ˆ k ) a † ˆ k, r i | i = X r =1 β ∗ Ar (ˆ k ) h | a ˆ p,s a † ˆ k, r | i = δ (ˆ k − ˆ p ) β ∗ As (ˆ k ) h | a ˆ p,s c † ˆ k,A | i = h | a ˆ p,s X r =1 h α Ar (ˆ k ) a † ˆ k, r + β Ar (ˆ k ) a ˆ k, r i | i = X r =1 α Ar (ˆ k ) h | a ˆ p,s a † ˆ k, r | i = δ (ˆ k − ˆ p ) α As (ˆ k )The latter quantity α As (ˆ k ) can thereof be interpreted as the relative probability amplitudethat a birefringent particle of mass m , frequency ω and wave vector ( k A , k , k ) and chiralpolarization vector ε µA (ˆ k ) is transmitted from the left face to the right face through thehyperplane x = 0 to become a Proca-St¨uckelberg particle with equal mass m , frequency ω and wave vector ( k , k , k ), but polarization vector e µs (ˆ k ). As an effect of this trans-mission, the first component of wave vector of a birefringent massive particle changes from k ± to k , while the longitudinal massive quanta do not change it’s wave vector.
4. CS Electrodynamics for a space-like CS vector
Let us consider in more details the case of a spatial Chern-Simons vector ζ µ = (0 , − ζ x , , ( (cid:3) + m ) A = − ζ x θ ( − x )( ∂ A − ∂ A )( (cid:3) + m ) A = 0( (cid:3) + m ) A = − ζ x θ ( − x )( ∂ A − ∂ A )( (cid:3) + m ) A = − ζ x θ ( − x )( ∂ A − ∂ A ) (4.1)– 14 –et’s introduce the vectors: ˆ k = ( ω, k , k ), ˆ x = ( x , x , x ), and their scalar productˆ k · ˆ x = − ωx + k x + k x . Using the Fourier transformation in these coordinates one cansolve the equation describing A in the entire space, A = Z d ˆ k (2 π ) θ ( ω − k ⊥ − m ) (˜ u → ( ω, k , k ) e ik x + ˜ u ← ( ω, k , k ) e − ik x ) e i ˆ k ˆ x (4.2)where k = ω − m − k ⊥ , k ⊥ = k + k .Now, let’s examine the remaining components of A µ . There are two solutions for eachcomponent in the different half-spaces. The first solution describes the vector meson physicsat x >
0, the second one is for x < x > A = R d ˆ k (2 π ) θ ( ω − k ⊥ − m ) (˜ u → ( ω, k , k ) e ik x + ˜ u ← ( ω, k , k ) e − ik x ) e i ˆ k ˆ x A = R d ˆ k (2 π ) θ ( ω − k ⊥ − m ) (˜ u → ( ω, k , k ) e ik x + ˜ u ← ( ω, k , k ) e − ik x ) e i ˆ k ˆ x A = R d ˆ k (2 π ) θ ( ω − k ⊥ − m ) (˜ u → ( ω, k , k ) e ik x + ˜ u ← ( ω, k , k ) e − ik x ) e i ˆ k ˆ x (4.3)To solve Eqs. (4.1) in the second case, we perform the Fourier transformation in x as wellon the entire axis, ( − ω + k + m ) ˜ A = − iζ x ( k ˜ A − k ˜ A )( − ω + k + m ) ˜ A = − iζ x ( k ˜ A + ω ˜ A )( − ω + k + m ) ˜ A = iζ x ( ω ˜ A + k ˜ A ) (4.4)This system leads to,˜ A ν = X A [˜ v νA → ( k , k , ω ) δ ( k − k A ) + ˜ v νA ← ( k , k , ω ) δ ( k + k A )] (4.5)Herein the first index of ˜ v denotes the corresponding component of A ν , ν = 0 , ,
3, thesecond index A stays for different mass-shell k for polarizations L, + , − and the arrows → , ← indicate the direction of particle propagation. The dispersion laws for differentpolarizations read, k L = k = q ω − m − k ⊥ k = r ω − m − k ⊥ + ζ x q ω − k ⊥ k − = r ω − m − k ⊥ − ζ x q ω − k ⊥ (4.6)– 15 –s well there are following relations between v , ˜ v ⇆ = k k − iω √ ω − k ⊥ ω − k ˜ v ⇆ ˜ v − ⇆ = k k + iω √ ω − k ⊥ ω − k ˜ v − ⇆ ˜ v ⇆ = − i ( k ˜ v ⇆ − k ˜ v ⇆ ) √ ω − k ⊥ ˜ v − ⇆ = i ( k ˜ v − ⇆ − k ˜ v − ⇆ ) √ ω − k ⊥ ˜ v L ⇆ = k k ˜ v L ⇆ ˜ v L ⇆ = − ωk ˜ v L ⇆ (4.7)Now we have the solutions in both parts of the space, and we have to match them on theboundary. For this purpose, let’s take the system (4.4) and perform Fourier transformation,using x , x , x . ( − ω + m + k ⊥ ) ˜ A − ∂ ˜ A = iζ x θ ( − x )( k ˜ A − k ˜ A )( − ω + m + k ⊥ ) ˜ A − ∂ ˜ A = − iζ x θ ( − x )( k ˜ A + ω ˜ A )( − ω + m + k ⊥ ) ˜ A − ∂ ˜ A = iζ x θ ( − x )( ω ˜ A + k ˜ A ) (4.8)This system is valid in all the space. To solve it we have to integrate over x in a smallvicinity of the boundary, from − ε to ε and compare the coefficients of the exponentialsin the left and right sides of the equations. The solutions( ν = 0 , ,
3) have the followinggeneral form,˜ A ν = ˜ u ν → ( ω, k , k ) e ik x + ˜ u ν ← ( ω, k , k ) e − ik x , x > P A (cid:2) ˜ v νA → ( ω, k , k ) e ik A x + ˜ v νA ← ( ω, k , k ) e − ik A x (cid:3) , x < − k ( − ˜ u → − ˜ u ← ik + P A ˜ v A → − ˜ v A ← ik A ) = iζ x ( k P A ˜ v A → − ˜ v A ← ik A − k P A ˜ v A → − ˜ v A ← ik A ) − k ( − ˜ u → − ˜ u ← ik + P A ˜ v A → − ˜ v A ← ik A ) = − iζ x ( k P A ˜ v A → − ˜ v A ← ik A + ω P A ˜ v A → − ˜ v A ← ik A ) − k ( − ˜ u → − ˜ u ← ik + P A ˜ v A → − ˜ v A ← ik A ) = iζ x ( ω P A ˜ v A → − ˜ v A ← ik A + k P A ˜ v A → − ˜ v A ← ik A ) (4.10)The contributions to amplitude in the right half-space ( u µ ) from the different dispersionlaws in the left half-space are independent: u µ = u ( L ) µ + u (+) µ + u ( − ) µ .If A = L : − k ( − ˜ u ( L )0 → − ˜ u ( L )0 ← ik + ˜ v L → − ˜ v L ← ik ) = iζ x ( k v L → − ˜ v L ← ik − k v L → − ˜ v L ← ik ) − k ( − ˜ u ( L )2 → − ˜ u ( L )2 ← ik + ˜ v L → − ˜ v L ← ik ) = − iζ x ( k v L → − ˜ v L ← ik + ω ˜ v L → − ˜ v L ← ik ) − k ( − ˜ u ( L )3 → − ˜ u ( L )3 ← ik + ˜ v L → − ˜ v L ← ik ) = iζ x ( ω ˜ v L → − ˜ v L ← ik + k v L → − ˜ v L ← ik )– 16 –t is easy to see that the right parts in this system are equal to zero (from (4.7)), so we get˜ u ( L ) ν → − ˜ u ( L ) ν ← = ˜ v νL → − ˜ v νL ← (4.11)If A = +: − k ( − ˜ u (+)0 → − ˜ u (+)0 ← ik + ˜ v → − ˜ v ← ik ) = iζ x ( k v → − ˜ v ← ik − k v → − ˜ v ← ik ) − k ( − ˜ u (+)2 → − ˜ u (+)2 ← ik + ˜ v → − ˜ v ← ik ) = − iζ x ( k v → − ˜ v ← ik + ω ˜ v → − ˜ v ← ik ) − k ( − ˜ u (+)3 → − ˜ u (+)3 ← ik + ˜ v → − ˜ v ← ik ) = iζ x ( ω ˜ v → − ˜ v ← ik + k v → − ˜ v ← ik )using the relations between v + (4.7): − k ( − ˜ u (+)0 → − ˜ u (+)0 ← ik + ˜ v → − ˜ v ← ik ) = (˜ v → − ˜ v ← ) ζ x √ ω + k ⊥ ik − k ( − ˜ u (+)2 → − ˜ u (+)2 ← ik + ˜ v → − ˜ v ← ik ) = (˜ v → − ˜ v ← ) ζ x √ ω + k ⊥ ik − k ( − ˜ u (+)3 → − ˜ u (+)3 ← ik + ˜ v → − ˜ v ← ik ) = (˜ v → − ˜ v ← ) ζ x √ ω + k ⊥ ik and for A = + we get: ˜ u (+) ν → − ˜ u (+) ν ← = (˜ v ν + → − ˜ v ν + ← ) k k (4.12)Finally, if A = − using the same method we obtain:˜ u ( − ) ν → − ˜ u ( − ) ν ← = (˜ v ν −→ − ˜ v ν −← ) k − k (4.13)Thus we have found the following matching conditions,˜ u ( L ) ν → − ˜ u ( L ) ν ← = ˜ v νL → − ˜ v νL ← ˜ u (+) ν → − ˜ u (+) ν ← = (˜ v ν + → − ˜ v ν + ← ) k k ˜ u ( − ) ν → − ˜ u ( − ) ν ← = (˜ v ν −→ − ˜ v ν −← ) k − k (4.14)In addition, all contributions of A are continuous.˜ u ( L ) ν → + ˜ u ( L ) ν ← = ˜ v νL → + ˜ v νL ← ˜ u (+) ν → + ˜ u (+) ν ← = ˜ v ν + → + ˜ v ν + ← ˜ u ( − ) ν → + ˜ u ( − ) ν ← = ˜ v ν −→ + ˜ v ν −← (4.15)From these sets of equations we obtain the relations between u and v . All of them can bewritten in such a form,˜ u ( A ) ν → = 12 (˜ v νA → ( k A + k k ) − ˜ v νA ← ( k A − k k )) (4.16)˜ u ( A ) ν ← = 12 ( − ˜ v νA → ( k A − k k ) + ˜ v νA ← ( k A + k k )) (4.17)– 17 – .2 Comparison of different representations Let’s compare our classical solutions for a space-like CS vector with the solutions obtainedby quantization formalism for the Bogolyubov transformation in Sec. 3. They must be thesame, consequently we have,˜ u ν → (2 π ) = X r =1 [ (2 π ) k ] − / a ˆ k,r e νr (ˆ k ) (4.18)˜ v νA → (2 π ) = [ (2 π ) k A ] − / c ˆ k,A ε νA (ˆ k ) (4.19)˜ v νA ← (2 π ) = [ (2 π ) k A ] − / c † ˆ k,A ε ν ∗ A (ˆ k ) (4.20)But we know that ˜ u ν → = P A ˜ u ( A ) ν → , so X r =1 [2 k ] − / a ˆ k,r e νr (ˆ k ) == X A
12 [2 k A ] − / (cid:20) c ˆ k,A ε νA (ˆ k ) ( k A + k k ) − c † ˆ k,A ε ν ∗ A (ˆ k ) ( k A − k k ) (cid:21) (4.21)Using the fact that − g µν e µr (ˆ k ) e νs (ˆ k ) = δ rs , a ˆ k,r = X A (cid:20) − c ˆ k,A g µν e µr (ˆ k ) ε νA (ˆ k ) k A + k √ k k A + c † ˆ k,A g µν e µr (ˆ k ) ε ν ∗ A (ˆ k ) k A − k √ k k A (cid:21) . (4.22)One can see that the same equation we obtained in Sec. 3 (3.22). Thus, our results areconsistent. Now we consider the case when the particle enters from the left half-space to the right one,so we have to take ˜ u µ ← = 0. ˜ u ( L ) ν → = ˜ v νL → − ˜ v νL ← (4.23)˜ u (+) ν → = (˜ v ν + → − ˜ v ν + ← ) k k (4.24)˜ u ( − ) ν → = (˜ v ν −→ − ˜ v ν −← ) k − k (4.25) ν = 0 , , u ( A ) ν → = ˜ v νA → + ˜ v νA ← (4.26)And now we can find which part is reflected: ˜ v νL ← = 0 (4.27)˜ v ν + ← = k − k k + k ˜ v ν + → (4.28)˜ v ν −← = k − − k k − + k ˜ v ν −→ (4.29)– 18 –nd which part is passed through: ˜ u ( L ) ν → = ˜ v νL → (4.30)˜ u (+) ν → = 2 k k + k ˜ v ν + → (4.31)˜ u ( − ) ν → = 2 k − k + k − ˜ v ν −→ (4.32)Finally, we have the solution consisting of:for x > A µ = Z d ˆ k (2 π ) θ ( ω − k ⊥ − m ) ˜ u µ → ( ω, k , k ) e ik x e i ˆ k ˆ x (4.33)and for x < A = R d ˆ k (2 π ) θ ( ω − k ⊥ − m ) ˜ u → ( ω, k , k ) e ik x e i ˆ k ˆ x A ν = R d ˆ k (2 π ) θ ( ω − k ⊥ − m ) P A (cid:2) ˜ v νA → ( ω, k , k ) e ik A x + ˜ v νA ← ( ω, k , k ) e − ik A x (cid:3) e i ˆ k ˆ x ν = 0 , , x < A ν = Z d ˆ k (2 π ) θ ( ω − k ⊥ − m ) X A h ˜ v νA → ( ω, k , k ) e − ikx + ˜ v νA ← ( − ω, − k , − k ) e ikx i (4.35) ν = 0 , , It is also interesting to look at the case of entering the parity breaking medium. It meansthat our particle moves from the left half-space to the right one. Thus we can take ˜ v µA → =0. From eqs. (4.17) one derives, ˜ u ( A ) ν → = k − k A k + k A ˜ u ( A ) ν ← (4.36)˜ v νA ← = 2 k k + k A ˜ u ( A ) ν ← (4.37)The solutions are: for x > A ν = Z d ˆ k (2 π ) θ ( ω − k ⊥ − m ) h ˜ u ν ← ( ω, k , k ) e − ik x + ˜ u ν → ( ω, k , k ) e ik x i e i ˆ k ˆ x (4.38)for x < A ν = Z d ˆ k (2 π ) θ ( ω − k ⊥ − m ) X A ˜ v νA ← ( ω, k , k ) e − ik A x e i ˆ k ˆ x ν = 0 , , u (+) ν ← , ˜ u ( − ) ν ← , ˜ u ( L ) ν ← . Wecan find them using (4.7), (4.36), (4.37) and all incoming amplitudes (˜ u ν ← ). Below arepresented the expressions for all components, ˜ u ( L )0 ← = ω ω − k ⊥ ˜ u ← + ωk ω − k ⊥ ˜ u ← + ωk ω − k ⊥ ˜ u ← ˜ u (+)0 ← = − k ⊥ ω − k ⊥ ) ˜ u ← − ωk − ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← − ωk + ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← ˜ u ( − )0 ← = − k ⊥ ω − k ⊥ ) ˜ u ← − ωk + ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← − ωk − ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← (4.40) ˜ u ( L )2 ← = − k ω − k ⊥ ˜ u ← − ωk ω − k ⊥ ˜ u ← − k k ω − k ⊥ ˜ u ← ˜ u (+)2 ← = ω − k ω − k ⊥ ) ˜ u ← + ωk − ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← + k k − iω √ ω − k ⊥ ω − k ⊥ ) ˜ u ← ˜ u ( − )2 ← = ω − k ω − k ⊥ ) ˜ u ← + ωk + ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← + k k + iω √ ω − k ⊥ ω − k ⊥ ) ˜ u ← (4.41) ˜ u ( L )3 ← = − k ω − k ⊥ ˜ u ← − ωk ω − k ⊥ ˜ u ← − k k ω − k ⊥ ˜ u ← ˜ u (+)3 ← = ω − k ω − k ⊥ ) ˜ u ← + ωk + ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← + k k + iω √ ω − k ⊥ ω − k ⊥ ) ˜ u ← ˜ u ( − )3 ← = ω − k ω − k ⊥ ) ˜ u ← + ωk − ik √ ω − k ⊥ ω − k ⊥ ) ˜ u ← + k k − iω √ ω − k ⊥ ω − k ⊥ ) ˜ u ← (4.42) Let’s build the Green’s function (the propagator) for each polarization separately. We cansay that our equations (4.8) have two linearly independent solutions, for example, one forthe particle moving from the left half-space, and another one for the particle moving fromthe right half-space. ψ ( A )1 = (cid:20) k A k + k A θ ( x ) e ik x + θ ( − x )( e ik A x + k A − k k A + k e − ik A x ) (cid:21) ˜ v νA → (4.43) ψ ( A )2 = (cid:20) k k + k A θ ( − x ) e − ik A x + θ ( x )( e − ik x − k A − k k A + k e ik x ) (cid:21) ˜ u ( A ) ν ← (4.44)Using the conventional Sturm-Liouville theory we can determine the Green function afterthe Fourier transformation in x , x , x , G ( A ) ( ω, k , k , x , x ′ ) = − W ( ψ ( A )1 , ψ ( A )2 ) ( ψ ( A )2 ( x ′ ) ψ ( A )1 ( x ) , x ≤ x ′ ψ ( A )1 ( x ′ ) ψ ( A )2 ( x ) , x > x ′ (4.45)The Wronskian is given by, W ( ψ ( A )1 ( x ′ ) , ψ ( A )2 ( x ′ )) = ( 4 ik k A k + k A )˜ u ( A ) ν ← ˜ v νA → , (4.46)Next let’s perform the Fourier transformation in x , x ′ .˜ G ( A ) ( ω, k , k , k , k ′ ) = Z Z ∞−∞ d x d x ′ i k + k A k k A ×× ( θ ( x ′ − x ) ψ ( A )2 ( x ′ ) ψ ( A )1 ( x ) + θ ( x − x ′ ) ψ ( A )1 ( x ′ ) ψ ( A )2 ( x )) e − ik x e − ik ′ x ′ =– 20 – ( i k + k A k k A ) { k A k + k A (cid:20) i v.p. k ′ + k + 1 i v.p. k + k (cid:21) (cid:20) i ( k ′ + k − iε ) (cid:21) −− k A ( k A − k )( k + k A ) (cid:20) i v.p. k ′ − k + 1 i v.p. k − k (cid:21) (cid:20) i ( k ′ + k − k − iε ) (cid:21) ++ 2 k k + k A (cid:20) i v.p. k ′ + k A + 1 i v.p. k + k A (cid:21) (cid:20) ik ′ + k + iε + k A − k k A + k ( ik ′ + k + 2 k A + iε ) (cid:21) ++ 2 k A k + k A (cid:20) πδ ( k ′ + k ) − k A − k k A + k πδ ( k ′ − k ) (cid:21) (cid:20) i ( k − k − iε ) (cid:21) ++ 2 k A k + k A (cid:20) πδ ( k + k ) − k A − k k A + k πδ ( k − k ) (cid:21) (cid:20) i ( k ′ − k − iε ) (cid:21) ++ (cid:20) i ( k ′ + k − iε ) − k A − k k A + k ( 1 i ( k ′ − k − iε ) ) − k k A + k i v.p. k ′ + k A (cid:21) ×× (cid:20) ik − k A + iε + k A − k k A + k ( ik + k A + iε ) (cid:21) ++ (cid:20) i ( k + k − iε ) − k A − k k A + k ( 1 i ( k − k − iε ) ) − k k A + k i v.p. k + k A (cid:21) ×× (cid:20) ik ′ − k A + iε + k A − k k A + k ( ik ′ + k A + iε ) (cid:21) } (4.47)
5. Conclusions and outlook
For massive photons and vector mesons the physical subspace consists of the three po-larizations A = ± , L , but the chiral transverse states with complex polarization vectors ε ν ± ( k ± ) propagate as massive states with effective masses dependent on the wave vector k ± . If we take the photon mass m γ = 0 one of helicity states becomes superluminal for allvalues of the momenta and produces a sort of Cherenkov radiation [75], gradually splittinginto three photons with negative polarizations (see [76, 77] for similar arguments for space-like background vectors). We have to stress that, kinematically, the high-energy photonwith positive polarization can also undergo splitting into the negative polarization photons.Both splittings are kinematically allowed as it can be easily read out from the inequalityfor the forward decay (we neglect here the photon mass), ω ± ( k ) = p k ± ζ | k | > ω − ( k . (5.1)Thus if the phenomenon of positron (dilepton) excess is accounted for by the instability ofphotons in a pseudoscalar background an accompanying effect might be the suppression ofhigh-energy γ rays from the same region, depending on the value of the effective photonmass, bearing in mind that this process is anyway a one-loop effect. In addition, thereis the possibility of “radiative” LIV decays e − → e − γ ; the momentum threshold being | k | > m e /ζ x . This effect will change the energy spectrum of the e + e − pair producedin LIV γ → e + e − decays, but it is suppressed by a power of α and the cross-sectionmust be proportional to ζ x too [51, 53, 54]. In spite of that axion density in galaxies isvery low there might be the regions of relatively dense axion localization (”axion stars”– 21 –49]) and to detect them one has to know how in-medium photons and leptons leave thoseregions. Thus boundary effects are crucial for their discovery. The influence of a boundarybetween parity-odd medium and vacuum on the decay width of photons and vector mesonsrepresents also a very interesting problem for calculation of realistic yield of dileptonsproduced by ρ, ω mesons in central heavy-ion collisions when local parity breaking occurs[62]. Thus it certainly deserves to be a subject of further investigation, in particular, ofquantization for time-like CS vector, of transmission and reflection of unstable photonsand vector bosons against a boundary and their relative dependence on energies, at last,of peculiarities of angular dependence allowing to detect their emission from axion cloudsor fireballs. Acknowledgments
This work is partially supported by Grants RFBR 09-02-00073-a and by SPbSU grant11.0.64.2010. A.A.A. acknowledges also the financial support from projects FPA2010-20807, 2009SGR502, CPAN (Consolider CSD2007-00042). S.S.K. is supported by DynastyFoundation stipend. We are grateful to Yu.M.Pis’mak for valuable comments and to P.Giacconi for her assistance at early stage of our work.
Appendix A
Let’s integrate eq.(4.9) from − ε to ε taking into account following relations: Z ε − ε d x ˜ A ν = Z − ε d x ˜ A ν + Z ε d x ˜ A ν Z ε − ε d x θ ( − x ) ˜ A ν = Z − ε d x ˜ A ν Z ε d x ˜ A ν = − ik (˜ u ν → − ˜ u ν ← ) + ˜ u ν → e ik ε ik + ˜ u ν ← e − ik ε − ik Z − ε d x ˜ A ν = X A [ 1 ik (˜ v νA → − ˜ v νA ← ) − ˜ v νA → e − ik ε ik − ˜ v νA ← e ik ε ik ] Z ε − ε d x ∂ ˜ A ν = ∂ ˜ A ν | ε − ε == ik ˜ u ν → e ik ε − ik ˜ u ν ← e − ik ε − X A [ ik A ˜ v νA → e − ik A ε − ik A ˜ v νA ← e ik A ε ] , and consider separately terms containing e ik A ε .Take now the first equation from the system (4.8) and prove the next equality:( − ω + m + k ⊥ ) ˜ v A ← ik A e ik A ε − ik A ˜ v A ← e ik A ε = iζ x ( k v A ← ik A e ik A ε − k v A ← ik A e ik A ε ) .To prove it, we multiply this equality by ik A :– 22 – k A + ( ω + m + k ⊥ ))˜ v A ← = iζ x ( k ˜ v A ← − k ˜ v A ← )Finally using expression for k A we get: ǫ A q ω − k ⊥ ˜ v A ← = i ( k ˜ v A ← − k ˜ v A ← ),where ǫ A = , A = + − , A = − , A = L and it coincides with relations (4.7). Similarly, one can check vanishing of terms with e − ik A ε and e ± ik ε . References [1] G. M. Shore, Nucl. Phys. B , 86 (2005) [arXiv:hep-th/0409125].[2] T. Jacobson, S. Liberati and D. Mattingly, Annals Phys. , 150 (2006)[arXiv:astro-ph/0505267].[3] J. Gamboa, J. Lopez-Sarrion and A. P. Polychronakos, Phys. Lett. B , 471 (2006)[arXiv:hep-ph/0510113].[4] K. Nozari and D. Sadatian, Electron. J. Theor. Phys. , 87 (2007).[5] V. A. Kosteleck´y and N. Russell, arXiv:0801.0287 [hep-ph].[6] W. Bietenholz, arXiv:0806.3713 [hep-ph].[7] F. W. Stecker and S. T. Scully, New J. Phys. , 085003 (2009) [arXiv:0906.1735[astro-ph.HE]].[8] J. Alfaro and P. Gonzalez, arXiv:0909.3883 [hep-ph].[9] R. Lehnert, J. Phys. Conf. Ser. , 012036 (2009) [arXiv:0907.1319 [hep-ph]].[10] L. Shao and B. Q. Ma, Mod. Phys. Lett. A , 3251 (2010) [arXiv:1007.2269 [hep-ph]].[11] S. M. Carroll, G. B. Field and R. Jackiw, Phys. Rev. D , 1231 (1990).[12] R. Jackiw, Int. J. Mod. Phys. B , 2011 (2000) [arXiv:hep-th/9903044].[13] G. E. Volovik, JETP Lett. , 1 (1999) [Pisma Zh. Eksp. Teor. Fiz. , 3 (1999)][arXiv:hep-th/9905008].[14] A.A. Andrianov, P. Giacconi and R. Soldati, JHEP , 030 (2002).[15] F. R. Klinkhamer and G. E. Volovik, Int. J. Mod. Phys. A , 2795 (2005)[arXiv:hep-th/0403037].[16] D. Ebert, V. C. Zhukovsky and A. S. Razumovsky, Phys. Rev. D , 025003 (2004)[arXiv:hep-th/0401241].[17] V. C. Zhukovsky, A. E. Lobanov and E. M. Murchikova, Phys. Rev. D , 065016 (2006)[arXiv:hep-ph/0510391].[18] J. Gamboa and J. Lopez-Sarrion, Phys. Rev. D , 067702 (2005) [arXiv:hep-th/0501034]. – 23 –
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