A quantum field model for tachyonic neutrinos with Lorentz symmetry breaking
aa r X i v : . [ h e p - ph ] A ug November 23, 2018 17:4 WSPC - Proceedings Trim Size: 9in x 6in ws-cpt10rev A QUANTUM FIELD MODEL FOR TACHYONICNEUTRINOS WITH LORENTZ SYMMETRY BREAKING
MAREK J. RADZIKOWSKI
Dept. of Physics and Astronomy, University of British ColumbiaVancouver, B.C. V6T 1Z1, CanadaE-mail: [email protected]
A quantum field model for Dirac-like tachyons respecting a frame-dependentinterpretation rule, and thus inherently breaking Lorentz invariance, is de-fined. It is shown how the usual paradoxa ascribed to tachyons, instability andacausality, are resolved in this model, and it is argued elsewhere that Lorentzsymmetry breaking is necessary to permit perturbative renormalizability andcausality. Elimination of negative-normed states results in only left-handed par-ticles and right-handed antiparticles, suitable for describing the neutrino. Inthis context the neutron beta decay spectrum is calculated near the end pointfor large, but not ultrarelativistic preferred frame speed, assuming a vectorweak interaction vertex.
1. Introduction
The purpose of this talk is very briefly to introduce a quantum field theo-retic model of Dirac-like tachyons (or
Dirachyons ), which must necessarilyincorporate spontaneous Lorentz symmetry breaking (LSB), a and to derivebest fit curves for beta decay, which experimentalists may use to test the tachyonic neutrino hypothesis . The metric signature is (+ , − , − , − ) and c = ~ = 1. The proper, orthochronous Lorentz group is denoted by L ↑ + . Anelement Λ ∈ L ↑ + is simply called a Lorentz transformation here, or LT forshort. b Figure 1 depicts the energy-momenta of (anti-)particles allowed in thismodel, in a typical inertial frame O . The spectrum is the upper half ofthe one-sheeted mass hyperboloid E − p = − m , sliced in two by thehyperplane E + βp z = 0. In the preferred tachyon frame T = O ′ , an observer a An analogous approach is offered by Ciborowski and Rembieli´nski. The LTs herein are considered to be observer or passive transformations. ovember 23, 2018 17:4 WSPC - Proceedings Trim Size: 9in x 6in ws-cpt10rev m = 1 , β = 0 . views this hyperplane as E ′ = 0. c Here, T ’s velocity relative to O has beenchosen to be − β ˆ z in O .Thus we have a frame-dependent interpretation rule . This is akin to the‘Reinterpretation Principle,’ by which a negative energy tachyon travellingbackward in time, is (classically) equivalent to a positive energy (anti-)tachyon moving forward in time, with opposite momentum, etc. However,in the QFT, instead of insisting that all tachyons have positive energiesin all frames, one henceforth requires this in T (mod SO(3)), and allowsnegative energies in all other frames, consistent with the action of L ↑ + ,without further reinterpretation, i.e., the corresponding positive energieswith opposite momenta are missing from the spectrum.
2. Definition of quantum field model
One seeks to quantize the equation, d originally due to Tanaka, ( i / ∂ − mγ ) ψ = 0 , (1)which is derivable from the Hermitian Lagrangian density L = i ¯ ψγ / ∂ψ − m ¯ ψψ . In O , an indefinite inner product is defined for solutions u ( x ) , u ( x ) c Henceforth an observer in T always uses primed coordinates. d A similar prescription works for scalar tachyons obeying the Klein-Gordon equationwith negative mass-squared term. ovember 23, 2018 17:4 WSPC - Proceedings Trim Size: 9in x 6in ws-cpt10rev of Eq. 1: ( u , u ) ≡ − Z t ′ = a ′ d x ′ u ′† ( x ′ ) γ u ′ ( x ′ ) . (2)Here, a Λ ∈ L ↑ + is chosen so that x = Λ x ′ , and u ′ ( x ′ ) , u ′ ( x ′ ) are foundso that u i ( x ) = D (Λ) u ′ i ( x ′ ) , i = 1 ,
2, where D is the (1 / , ⊕ (0 , / L ↑ + , i.e., ( u , u ) is invariant under observer LTs. Adheringto the above interpretation rule, the ansatz for ˆ ψ ( x ) in O isˆ ψ ( x ) = X s Z p ∈ U dν ( p ) (cid:2) u p,s ( x ) a p,s + v p,s ( x ) b † p,s (cid:3) , (3)where dν ( p ) is the invariant measure on the mass hyperboloid. Also, u p,s ( x ) [ v p,s ( x )] are ‘upper’ [‘lower’] energy solutions of Eq. (1) of the formexp( − ip · x ) u s ( p ), [exp( ip · x ) v s ( p )], for p ∈ U , obtained from suitablynormalized positive [negative] energy solutions of ( p / ′ − mγ ) u ′ s ( p ′ ) = 0,[( p / ′ + mγ ) v ′ s ( p ′ ) = 0] in T via D (Λ). The parameter s = ± denoteshelicity eigenstates. The creation/annihilation operators a p,s , b p,s , a † p,s , b † p,s satisfy the anticommutation relations n a p,s , a † q,s ′ o = n b p,s , b † q,s ′ o = ± E p δ ss ′ δ (3) ( p − q ) , (4)and all other anticommutators vanish. Here, E p ≡ p p − m , | p | ≥ m , andthe sign is negative for the negative normed modes. Since the anticommu-tators are positive definite, one must eliminate such modes from the fieldoperator, yieldingˆ ψ ( x ) = Z p ∈ U dν ( p ) h u p, − ( x ) a p, − + v p, + ( x ) b † p, + i . (5)The vacuum state | i O is defined by a p,s | i O = b p,s | i O = 0, for all p ∈ U .The subscript is included since this vacuum is not L ↑ + invariant. The n -point functions (expectation values of products of field operators which inturn define a QFT ), and Green’s functions may then be computed in O .With the appropriate sign in front of the classical Hamiltonian density, thesecond quantized Hamiltonian (defined analogously to Eq. (2)) becomesˆ H = Z p ∈ U dν ( p ) p (cid:16) a † p, − a p, − + b † p, + b p, + (cid:17) , (6)which is positive semidefinite only in T . Furthermore, there remains a con-sistent notion of lepton number, namely the quantization of the inner prod-uct, Eq. (2), ˆ Q = Z p ∈ U dν ( p ) (cid:16) a † p, − a p, − − b † p, + b p, + (cid:17) . (7) ovember 23, 2018 17:4 WSPC - Proceedings Trim Size: 9in x 6in ws-cpt10rev
3. Resolution of paradoxa
The usual difficulties ascribed to tachyons are instability (of the vacuum)and acausality. There are two possible kinds of instability: (1) due to ex-ponentially growing/decaying modes in time, and (2) due to negative realenergies unbounded from below, leading to lack of perturbative renormal-izability. Instabilities of type (1) are excluded from this model by fiat, sincethey would correspond to unphysical imaginary energies. e An instabilityof type (2) is avoided due to the spectral cutoff. Note that requiring theusual Lorentz covariance of the two point function would lead to a non-Hadamard state having an ill-defined stress-energy tensor, while utilizingthe cutoff is conjectured to permit renormalizibility when the free model isincorporated into an interacting theory. The cutoff also prevents the cre-ation of causality-violating devices, such as ‘anti-telephones.’ Furthermore,the usual connection of spin and statistics is maintained in this model. Thisdiffers from Feinberg, who evidently ruled out commutation relations forthe scalar case due to a sign problem that can be traced back to the use ofan inappropriate surface of integration with which to evaluate inner prod-ucts. In the present case, one desires a regular massless model with brokenparity in the limit as m →
0, which would require the usual spin-statisticsconnection to hold for this tachyonic model.
4. Results
To evaluate the rate for neutron beta decay ( n −→ p + e + ¯ ν ), one re-places the Feynman diagram vertex factor − i ( g W / √ − γ ) / γ µ by − i ( g W / √ γ µ , since parity breaking is already accounted for in the freemodel. With neutron, antineutrino, proton and electron momenta denoted p , p , p , p resp., one arrives at the manifestly positive expression D |M| E = 2 (cid:18) g W M W (cid:19) [( p · p )(˜ p · p ) + ( p · p )(˜ p · p ) − m n m p (˜ p · p ) − m ν m e ( p · p ) + 2 m n m p m ν m e ] . (8)In preferred frame coordinates, ˜ p ′ ≡ ( | p ′ | , ( p ′ / | p ′ | ) p ′ ) is the future time-like conjugate of the spacelike p ′ = ( p ′ , p ′ ). To find ˜ p in any other frame O ,transform the coordinates of p to T , take the timelike conjugate, then trans-form back to O . With γ − = p − β , ∆ ≡ m n − m p , x ≡ (∆ − E ) /m ν , e These could not be excited perturbatively from real energies, since they would requirea four-point interaction vertex. ovember 23, 2018 17:4 WSPC - Proceedings Trim Size: 9in x 6in ws-cpt10rev and p e ≡ p ∆ − m e , and assuming the detected electrons are emitted in acone of half-angle 90 ◦ whose axis is at polar angle α , the differential decayrate is d Γ dE = m ν π (cid:18) g W M W (cid:19) p e (cid:26) θ ( x − βγ ) (cid:20) m e p x + 1 + 2 (cid:18) ∆ · x + p e β cos α (cid:19) + 1 β (cid:18) ∆ − p e β (1 + β ) cos α (cid:19) ln (cid:18) β − β (cid:19)(cid:21) + θ ( βγ − x ) θ ( x + βγ ) 1 β h m e ( x + β p x + 1) + β ∆ · x + p e α + x p x + 1 (cid:18) ∆ − p e βγ cos α (cid:19) − γ (cid:18) ∆ − p e β cos α (cid:19) x + (cid:18) ∆ − p e β (1 + β ) cos α (cid:19) ln (cid:16) γ (1 + β )( x + p x + 1) (cid:17)(cid:21)(cid:27) . (9)In the approximations leading to the above, the preferred frame speed β isallowed to be large, i.e., of order 1, but not ultrarelativistic. An appropri-ately modified set of fit curves may be used to test for tachyonic neutrinosand consequent Lorentz symmety breaking, e.g., at KATRIN. Acknowledgments
The author wishes to thank A.S. Wightman, K. Fredenhagen, G. Heinzel-mann, B.S. Kay, W.G. Unruh, and V.A. Kosteleck´y for discussions, hospi-tality and encouragement. Partial support was provided through a DFG atthe II. Institut f¨ur Theoretische Physik, Univ. of Hamburg, Germany, andthrough a teaching post-doctoral fellowship with PHAS at UBC.
References
1. J. Ciborowski and J. Rembieli´nski, in Z. Ajduk and A.K. Wr´oblewski, eds.,
Proceedings of the 28th International Conference on High Energy Physics(ICHEP’96),
World Scientific, Singapore, 1997, p. 1247.2. A. Chodos, A.I. Hauser, and V.A. Kosteleck´y, Phys. Lett. B , 431 (1985).3. O.M.P. Bilaniuk, V.K. Deshpande, and E.C.G. Sudarshan, Am. J. Phys. ,718 (1962).4. S. Tanaka, Progress of Theoretical Physics , 171 (1960).5. A.S. Wightman, Phys. Rev. , 860 (1956).6. M.J. Radzikowski, arXiv:0804.4534v2.7. G. Feinberg, Phys. Rev. , 1089 (1967).8. KATRIN Collaboration, A. Osipowicz et al.et al.