Neutrino orbital angular momentum in a plasma vortex
aa r X i v : . [ phy s i c s . p l a s m - ph ] A p r Neutrino orbital angular momentum in a plasma vortex
J. T. Mendonc¸a ∗ IPFN and CFIF, Instituto Superior T´ecnico, 1049-001 Lisboa, Portugal
B. Thid´e † Swedish Institute of Space Physics, P. O. Box 537, SE-751 21 Uppsala, Sweden
It is shown that an electron-neutrino beam, propagating in a background plasma, can be decomposed intoorbital momentum (OAM) states, similar to the OAM photon states. Coupling between different OAM neutrinostates, in the presence of a plasma vortex, is considered. We show that plasma vorticity can be transfered to theneutrino beam, which is relevant to the understanding of the neutrino sources in astrophysics. Observation ofneutrino OAM states could eventually become possible.
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I. INTRODUCTION
Neutrino beam interaction with a dense plasma has receivedconsiderable attention in recent years. In particular, it was rec-ognized that collective neutrino-plasma displays strong simi-larities with laser-plasma interaction phenomena. This is dueto the fact that neutrino dispersion relation in a plasma is for-mally analogous to the photon dispersion relation [1]. Cou-pling between neutrino beams and dense plasmas can then beconceived [2, 3], which is particularly relevant for the under-standing of supernova type II explosions, where an intenseand short neutrino burst is emitted [4]. It was also recognizedthat neutrino ponderomotive force can excite kinetic plasmainstabilities [5, 6], resulting from negative neutrino Landaudamping of electron plasma waves, similar to that observedfor photons [7]. Furthermore, it is also possible to show thatneutrinos acquire an effective electric charge in a plasma, ina way very similar to photons [8, 9]. Another important as-pect is related to the possible excitation of relativistic plasmawakefields by a neutrino burst [10, 11], and the generation ofmagnetic fields [12], through processes that mimic those ofphoton beams in a plasma.In this work, the similarities between neutrino and photondispersion are explored even further, by introducing the con-cept of neutrino orbital angular momentum (OAM). This is anew aspect of neutrino plasma physics, which is related withthe possible exchange of orbital angular momentum between aneutrino beam and a rotating plasma. It is known that photonscan carry, not only intrinsic angular momentum or spin, whichis associated with their polarization state, but also orbital an-gular momentum (OAM) [13]. The existence of OAM pho-ton states has been experimentally demonstrated early in 1936[14], but only recently photon OAM momentum received at-tention, after the demonstration that Laguerre-Guassian lasermodes correspond to well defined OAM modes, and that thesephoton modes not only can be measured as a photon beamproperty [15, 16], but can also be detected at the single pho- ∗ Electronic address: [email protected] † Also at LOIS Space Centre, V¨axj¨o University, SE-351 95 V¨axj¨o, Sweden ton level [17].Utilization of photon OAM in the low frequency radio wavedomain was recently proposed by one of us [18], as an ad-ditional method for studying the properties of radio sources.Here we show that a neutrino beam can be decomposed intoOAM momentum states, similar to the OAM photon states,and we study the coupling between these modes, when theneutrino beam propagates across a plasma vortex region. Wewill also show that plasma vorticity can be transferred to theneutrino beam, which can likewise be of great relevance to theunderstanding of the astrophysical sources of neutrino emis-sion. Detection of finite neutrino OAM states would give usadditional information on the collapsing star, in the case ofa supernova explosion, and would allow us to measure thevortex properties of the surrounding dense plasma. Observa-tion of neutrino OAM states could eventually be possible, bystudying the conservation of total angular momentum at de-tection, and by adapting the existing detection schemes to thatpurpose. Detection of radio signals from neutrinos interactingwith the Moon, as recently proposed [19], could provide onepossible method for measur ing such neutrino OAM states.
II. BASIC EQUATIONS
We know that the spinor field y describing electron-neutrinos moving in a dense plasma can be described by theDirac equation i ¯ h ¶¶ t y = H y , H = b m n c + a · p c + V (1)where m n is a non-zero rest mass, b and a are the well knownoperators. For simplicity, the plasma medium is assumedisotropic, and magnetic field effects are ignored. We retainexplicit physical units, with c = h =
1, for future com-parison between neutrino and photon states in a plasma. Theeffective plasma potential V ( r , t ) , results from the weak inter-actions between the neutrinos and the background electronsand ions (or protons) of the background plasma. We knowthat, for an electrically neutral medium, the proton contri-butions are exactly canceled by the neutral part of the elec-tron contribution, and the resulting plasma potential is simplygiven by V = √ G F n e ( r , t ) = ¯ hg w p , g = √ e m e e ¯ h G F (2)where n e is the electron mean plasma density, w p is theelectron plasma frequency, e and m e are the electron chargeand mass, e is the permittivity of vacuum, and G F is theFermi constant. Equations (1)-(2) have been used to study thecollective decay of plasmons into neutrino-antineutrino pairs[20, 21]. Here, we neglect such neutrino-antineutrino cou-pling processes. We also neglect mass oscillations, but theirpossible contribution will be discussed later. For a constantplasma density, we can use the Foldy-Wouthuysen procedure,leading to a diagonalized Hamiltonian H = b q m n c + p c + V (3)We will focus on a single spin (or helicity, in the limit of anegligible mass) electron-neutrino state. The correspondingsolution can then be written as y ( r , t ) = Y exp ( i k · r − i w t ) ,where the neutrino frequency w = < H > / ¯ h , and the wavevec-tor k = < p > / ¯ h satisfy the neutrino dispersion relation ( w − g w p ) = k c + w (4)with w = m n c / ¯ h . This shows a strong similarity with thephoton dispersion relation in a plasma. Let us now considera non-homogeneous and non-stationary medium, where theplasma density and the plasma potential are no longer con-stant, but vary on time a length scales much longer than theneutrino period and wavelength. V ( r , t ) = V + ˜ V ( r , t ) (5)which is the consequence of a space-time variation of thebackground plasma density and of the corresponding plasmafrequency, as described by w p = w p [ + e ( r , t )] = w p (cid:20) + n ˜ n ( r , t ) (cid:21) (6)where ˜ n is the electron density perturbation. By comparingthe two expressions, we conclude that˜ V ( r , t ) = ¯ hg w p e ( r , t ) = ¯ hg w p n ˜ n ( r , t ) (7)In order to solve the neutrino field equation, we can use thequasi-classical or WKB approximation and make the replace-ments H → ¯ h (cid:18) w − i ¶¶ t (cid:19) , p → ¯ h ( k + i (cid:209) ) . (8)The Hamiltonian operator (3) then becomes (cid:18) w − i ¶¶ t (cid:19) = b q w + ( k + i (cid:209) ) c + h ( V + ˜ V ) (9) Assuming that the dispersion relation (4) is always locally sat-isfied for the non-perturbed plasma potential V , we can use aWKB solution of the form y ( r , t ) = Y ( r , t ) exp ( i k · r − i w t ) (10)where Y ( r , t ) is now a slowly varying amplitude. Expandingthe operator (8) we can then write the evolution equation forthis amplitude as − i ¯ h ¶¶ t Y = c ( w − g w p ) " i k · (cid:209) − (cid:209) − c ( k · (cid:209) ) ( w − g w p ) + ˜ V Y (11)Noting that the neutrino mean velocity can be identified withthe group velocity associated with the dispersion relation (4),we get v n = ¶w¶ k = c k w − g w p (12)We can also introduce the neutrino relativistic factor g n , asdefined by m n g n c = ¯ h ( w − g w p ) (13)Using these quantities, we can rewrite equation (11) in a moreappropriate form i (cid:18) ¶¶ t + v n · (cid:209) (cid:19) Y = ¯ h m n g n " (cid:209) + (cid:16) v n c · (cid:209) (cid:17) + g w p n ˜ n Y (14)In the limit of a negligible neutrino velocity, this would takethe form of a Schroedinger equation. A similar equation wasalso derived for the case of a quantum free electron laser [22,23]. Let us first assume that a neutrino beam propagates alonga given axis Oz , in the absence of any plasma perturbation, ˜ n =
0. Retaining only the first order derivatives in the z- direction,we can then reduce the wave equation to a paraxial equation,similar to that describing a laser beam in the focal region (cid:18) (cid:209) ⊥ + ik ¶y¶ z (cid:19) y = k obeys the dispersion relation (4). Wecan try a Gaussian solution of the form y ( r ) ≡ y ( r , z ) = Y ( z ) exp (cid:20) ik r R ( z ) (cid:21) (16)Using the transverse Laplacian in cylindrical coordinates, werealize that this solution satisfies the paraxial equation, if weassume that R ( z ) = R + ( z − z ) , Y ( z ) = u R R ( z ) (17)where R and u are constants. Very simple arguments can beused to justify the relevance of such Gaussian beam solutionsin neutrino physics. For instance, the intense neutrino burstemitted by a collapsing type II supernova, is naturally focusedby the plasma inhomogeneities of the star material. The neu-trino dispersion relation (4) shows that neutrinos will be fo-cused into the lower density plasma regions, in the same wayas high energy photons would do. Another reason for neutrinofocusing into a nearly Gaussian form, would be the occurrenceof filamentational instability of the ultra intense neutrino burstcoming out of the collapsing star. A more general cylindricaltype of solution of the paraxial equation (15) can be repre-sented in the basis of orthogonal Laguerre-Gaussian modes,as defined by y ( r , t ) = (cid:229) pl F pl F pl ( r , f ) exp ( ikz − i w t ) (18)with F pl ( r , f ) (cid:181) (cid:18) r w (cid:19) | l | L | l | p (cid:18) r w (cid:19) exp ( il f − r w ) (19)The relevance of this type of representation will become ap-parent below. We recover the particular Gaussian solution bytaking l =
0, and R = − iw . The new quantity w defines theneutrino beam waist. By making an appropriate choice of thenormalization factors multiplying equation (19), we can verifythe orthogonality conditions Z ¥ rdr Z p d f F ∗ pl ( r , f ) F p ′ l ′ ( r , f ) = d pp ′ d ll ′ (20) III. NEUTRINO MODE COUPLING
Let us now consider the case where the electron density andthe plasma potential are not constant, ˜ n =
0. In cylindricalgeometry, we can use a general representation in terms of theLaguerre-Gaussian modes, as˜ n ( r , t ) = (cid:229) l Z dq p Z d W p ˜ n l ( r , q , W ) exp ( il f + iqz − i W t ) (21)To be more specific, we can concentrate on helical type ofperturbations described by˜ n ( r , t ) = ˜ n ( r ) exp (cid:18) − r a (cid:19) cos ( l f + iqz − i W t ) (22)In the presence of such density perturbations, the differentLaguerre-Gaussian modes of the neutrino field will be cou-pled, and the generic wavefunction solution will take the form y ( r , t ) = (cid:229) n , p , l y npl ( r ) exp [ iS ( r , t )] (23)where y npl are slowly varying ampitudes, and S ( r , t ) is thephase function as defined by S ( r , t ) = k n z − w n t . Here we haveused w n = w + n W , and the corresponding wavenumbers k n are determined by the neutrino dispersion, with w replaced by w n , valid for the unperturbed plasma density n . We can use,for the above amplitudes y npl ( r ) = Y npl ( z ) F pl ( r , f ) (24) Replacing this in the paraxial equation, where the term corre-sponding to the perturbed potential is retained, we obtain thefollowing mode coupling equations ddz Y npl ( z ) = − i (cid:229) n ′ p ′ l ′ K ( npl , n ′ p ′ l ′ ) Y n ′ p ′ l ′ ( z ) exp ( i D nn ′ z ) (25)where we have defined the wavenumber mismatch D nn ′ =( k n ′ − k n ± q ) , and used the following coupling coefficients K ( npl , n ′ p ′ l ′ ) = g w n k n c w p n d ( n − n ′ ± ) × Z p d f Z ¥ rdr ˜ n ( r ) F p ′ l ′ ( r ) F ∗ pl ( r ) exp [ i ( l ′ ± l − l ) f ] (26)Here, in the expressions of F ∗ pl ( r ) and F p ′ l ′ ( r ) we have takenout the factors exp ( − il f ) and exp ( il ′ f ) . The coupled modeequation (25) shows that, if we start from an initial neutrinostate with no orbital angular momentum, l ′ =
0, we will excitestates of finite angular momentum l =
0, due to the existenceof a perturbed plasma helical structure. But such coupling hasto verify certain resonant conditions. Depending on the radialprofile of the plasma vortex ˜ n ( r ) , various modes p = p ′ can beexcited. But we take here the simplest case of p = p ′ , whichallows us to drop this index from the coupled mode equations,and write them in a much simpler form ddz Y n ( z ) = − iK h Y n + ( z ) e i D z + Y n − ( z ) e − i D z i (27)Here n is an integer, and we have used a simplified notationfor the amplitudes, Y n ≡ Y n + n , p , l + n l . Approximate expres-sions for the coupling coefficients and wavenumber mismatchcan be written as K ≃ g w kc w p n ˜ n ( ) , D ≃ (cid:18) W c + q (cid:19) (28)For an exact phase matching, such that q = − W / c , the cou-pled mode equations (27) can easily be integrated in termsof Bessel functions. For initial conditions such that Y l ( ) = Y d l , corresponding to an initial Gaussian neutrino beam, weget the following amplitude for the various interacting modes Y n ( z ) = i n Y J n ( Kz ) (29)This shows that the states of higher neutrino OAM can be pop-ulated by a linear cascading process, approximately describedby this simple law. Such a simple but physically relevant solu-tion is valid for arbitrary interacting distances in the case of aperfect phase matching D =
0. And it remains valid for a finitephase mismatch, for interaction distances much shorter that2 p / D . However, in this more general case and for arbitrarilylong distances, the transfer of OAM from the plasma rotat-ing vortex to the neutrino beam will eventually be blocked byphase mixing. IV. CONCLUSIONS
In this work we have considered the orbital angular mo-mentum of a neutrino beam in a plasma, to our knowledge forthe first time. We have studied the interaction of the neutrinobeam with a plasma vortex, and explored the similarities be-tween neutrino and photon dispersion in a plasma. We haveshown that a neutrino beam, propagating in a dense plasmabackground, can be generally described by a superposition ofOAM states, similar to the photon OAM states recently ex-plored in the literature. The existence of plasma vortices in-troduces coupling between different neutrino OAM states. Wehave shown that plasma vorticity can be transfered to the neu-trino beam. This process can be of particular relevance to theunderstanding of the astrophysical neutrino sources. Studyof total angular momentum conservation, using the existingneutrino detection systems, could eventually lead to the first observation of neutrino OAM states.In this work, a simplified neutrino description was used.In particular, we have neglected the neutrino mass oscillationprocess. It is known that in a dense plasma, these oscilla-tions are resonantly enhanced by the so called MSW effect.This means that the existence of plasma vorticity in the res-onant plasma region where the MSW effect takes place willbe able to excite OAM states of the different neutrino species.Generalization of the MSW effect, taking the OAM states intoaccount, is in progress and will be the subject of a future work. [1] H.A. Bethe, Phys.Rev.Lett., , 1305 (1986).[2] R. Bingham, J.M. Dawson, J.J. Su and H.A. Bethe, Phys Lett.A, , 279 (1994).[3] J.T. Mendonc¸a, R. Bingham, P.K. Shukla, J.M. Dawson andV.N. Tsytovich, Phys.Lett.A, , 78 (1995).[4] H.A. Bethe and J.R. Wilson, Astrophys.J., , 14 (1985).[5] L.O. Silva, R. Bingham, J.M. Dawson, J.T. Mendonc¸a and P.K.Shukla, Phys.Rev.Lett., , 2703 (1999).[6] V.N. Oraevsky and V.B. Semikoz, PhysicaA, , 135 (1987).[7] R. Bingham, J.T. Mendonc¸a and J.M. Dawson, Phys.Rev.Lett., , 247 (1997).[8] A. Serbeto, L.A. Rios, J.T. Mendonc¸a and P.K. Shukla, Phys.Plasmas, , 1352 (2004).[9] J.T. Mendonc¸a, A. Serbeto, P.K. Shukla and L.O. Silva, Phys.Lett.B, , 63 (2002).[10] P.K. Shukla, R. Bingham, J.T. Mendonc¸a and L. Stenflo, Phys.Plasmas, , 1 (1998).[11] A. Serbeto, L.A. Rios, J.T. Mendonc¸a, P.K. Shukla and R. Bing-ham, JETP, , 466 (2004).[12] P.K. Shukla, L. Stenflo, R. Bingham, H.A. Bethe, J.M. Dawsonand J.T. Mendonc¸a, Phys.Plasmas, , 2815 (1998). [13] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw and J.P. Weerd-man, Phys.Rev.A, , 8185 (1992).[14] R.A. Beth, Phys.Rev., , 115 (1936).[15] M. Harris, C.A. Hill, P.R. Tapster and J.M. Vaughan, Phys.Rev.A, , 3119 (1996).[16] M.J. Padgett, J. Arlt, N.B. Simpson and L. Allen, Am.J.Phys., , 77 (1996).[17] J. Leach, M.J. Padgett, S.M. Barnett, S. Franke-Arnold and J.Courtial, Phys.Rev.Lett., , 257901 (2002).[18] B. Thid´e, H. Then, J. Sj¨oholm, K. Palmer, J. Bergman, T. D.Carozzi, Ya. N. Istomin, N. H. Ibragimov and R. Khamitova,Phys.Rev.Lett., , 087701 (2007).[19] O. St˚al, J.E.S. Bergman, B. Thid´e, L.K.S. Daldorff and G. In-gelman, Phys.Rev.Lett., , 071103 (2007).[20] A. Serbeto, Phys.Lett.A, , 217 (2002).[21] J.T.Mendonc¸a, A. Serbeto, R. Bingham and P.K. Shukla, J.PlasmaPhys., , 119 (2005).[22] A. Serbeto, J.T. Mendonc¸a, K.H. Tsui, and R. Bonifacio, Phys.Plasmas, , 013110 (2008).[23] G. Preparata, Phys.Rev.A,38