Classical neutral point particle in linearly polarized EM plane wave field
CClassical neutral point particle in linearly polarized EM plane wave field
Martin Formanek, ∗ Andrew Steinmetz, and Johann Rafelski
Department of PhysicsUniversity of ArizonaTucson, AZ, 85719, USA (Dated: April 25, 2019)We study a covariant classical model of neutral point particles with magnetic moment interact-ing with external electromagnetic fields. Classical dynamical equations which reproduce a correctbehavior in the non-relativistic limit are introduced. We also discuss the non-uniqueness of the co-variant torque equation. The focus of this work is on Dirac neutrino beam control. We present a fullanalytical solution of the dynamical equations for a neutral point particle motion in the presence ofan external linearly polarized EM plane wave (laser) fields. Neutrino beam control using extremelyintense laser fields could possibly demonstrate Dirac nature of the neutrino. However, for linearlypolarized ideal laser waves we show cancellation of all leading beam control effects.
PACS numbers:
I. INTRODUCTION
Since neutral particles (such as neutrons or Dirac neu-trinos) posses a magnetic moment they experience aStern-Gerlach type force when exposed to external fields.We invoke a theoretical framework developed for chargedparticles in our previous paper [1].In a first look on the environment of the neutral parti-cle – laser interactions using our framework [2] we focusedon describing the acceleration force neutral particles ex-perience. We found out that the square root of invariantacceleration is greatly enhanced by the relativistic factor γ with which the particle enters the field. This moti-vated our present study in which we aim to describe themotion of the neutral particles by solving the dynamicalequations.Classical solution for the charged particle dynamicsin a linearly polarized external plane wave field is wellknown [3–5]. The objective of our present work is toexpand on this knowledge by describing how this EMfield configuration affects neutral particles with magneticmoment. In doing so we employ mathematical methodsdeveloped in context of solution of the Landau-Lifshitzequation for charged particles in an external plane wavefield [6, 7]. Specifically we adapt technique of projectingthe 4-velocity and spin 4-vector on the wave vector andpolarization vector of the field and solving differentialequations for these projections.As our application we focus on laser interactions withultra-relativistic neutrinos. Neutrinos remain the leastunderstood elementary particles [8] and there is currentintense interest in furthering comprehension of their ele-mentary properties. Perhaps the most elementary ques-tion about the neutrino is if it is a Dirac or Majoranaparticle. Today, there is intense interest in search fordouble beta decay [9] with new experiments planned [10]which could prove that neutrinos are Majorana particles. ∗ [email protected] On the other hand we cannot dismiss possibility thatneutrinos are Dirac particles. A minimal extension ofthe Standard model [11] gives us a lower bound for themagnetic moment of the Dirac neutrino µ i ≈ . × − (cid:16) m i eV (cid:17) µ B , (1)where µ B = e (cid:126) / m e is the Bohr magneton. The m i isthe neutrino mass eigenstate ν i , which for electron neu-trino has scale m ν ≤ . µ ν < . × − µ B . (2)Detection of neutrino magnetic moment in this rangewould be complementary to a (so far nill) result of thesearch for double beta decay since it is believed that aMajorana neutrino just like a photon cannot have a mag-netic moment. Any experiment exploiting and detectingneutrino magnetic moment would therefore proof theirDirac nature.This study was motivated by the possibility of explor-ing control of neutrino beams by ultra-laser intense laserfields. Having such capabilities would be of an utmost im-portance because manipulating neutrino beams througha well defined interaction could resolve the question ofDirac/Majorana character of neutrinos and could pro-vide an opportunity to directly measure their mass andmagnetic moment.At the present moment, there is a multitude of com-peting models which approach relativistic dynamics of apoint particle from first principles. Thomas and Frenkel[13, 14] introduced in 1926 Frenkel’s equation of motionfor spin as a second rank tensor. Bargmann, Michel,and Telegdi formulated in late 1950s TBMT equations[15] which are until today often used for classical de-scription of the spin dynamics despite missing the Stern-Gerlach-like force. Another formulation is looking on a r X i v : . [ h e p - ph ] A p r a classical limit of the relativistic quantum-mechanicalDirac equation using the Foldy-Wouthuysen transforma-tion [16]. Unfortunately, the generalization of relativisticquantum description for particles with anomalous mag-netic moment is still not clear [17]. An overview of theseapproaches and their numerical tests can be found in[18, 19]. Our model [1] expands on the pioneering workof the first formulations. We incorporate force on a mag-netic dipole to the TBMT equations with an arbitraryanomalous magnetic moment.The interaction between neutrinos and external planewave field can be also treated within bounds of quantumfield theory. This has been shown in [20] for a specificmodel of anomalous magnetic moment/electric dipole in-teraction to describe decay γ → ν ¯ ν , and bremsstrahlungprocesses. We are interested in the classical behavior, be-cause except for the relic neutrinos, the neutrinos whichwe typically encounter (from nuclear reactions in theSun, our nuclear reactors, radioactive decay in Earth orsources outside of are solar system) are ultra-relativisticand within the classical limit for visible laser light λ ν λ γ (cid:28) , (3)where λ ν and λ γ are de Broglie wavelength of a neutrinowith a given energy and wavelength of the light respec-tively.Our paper is organized in three sections. In section IIwe present the dynamical equations for the neutral par-ticles and provide a rationale why we have chosen theirparticular form. In the section III we study a specific caseof external linearly polarized plane wave field. In sectionIV we discuss the case of ultra-relativistic neutrinos.Notation remark: we use the following convention forthe metric tensor g µν and totally anti-symmetric covari-ant pseudo-tensor (cid:15) µναβ diag g µν = { , − , − , − } , (cid:15) = − (cid:15) = +1 , (4)also SI units are used throughout. II. FORMULATION OF NEUTRAL PARTICLEDYNAMICS
In the paper [1] we introduced a generalized LorentzForce equation which allows us to account for a Stern-Gerlach force acting on a magnetic moment of point par-ticle in the presence of external electromagnetic fields˙ u µ = 1 m ( eF µν − ds · ∂F ∗ µν ) u ν , (5)where d is the constant of proportionality between parti-cle spin and magnetic moment | µµµ | = cd | sss | , µµµ = µ sss | sss | , (6) superscript ∗ denotes a dual tensor F ∗ µν = 12 (cid:15) µναβ F αβ , (7)and a ‘dot’ means a derivative with respect to proper time τ . As was discussed in [1] the form of the correspondingdynamical torque equation for spin is not unique. Similarnon-uniqueness manifests itself also in the quantum case[17] where extensions to Dirac and Klein-Gordon Pauliequations to accommodate magnetic moment differ sub-stantially. In the classical case we can only demand thatthe spin dynamics is consistent with the particle motiondynamics u · s = 0 , (8)and in the instantaneous frame co-moving with the par-ticle the equation for the spin dynamics has to contain atorque term dsssdt = µµµ × BBB , (9)which ensures the correct torque behavior when magneticmoment of particle is trying to align itself with the ex-ternal field [21]. This observation discussed before forcharged particles has to be true for a neutral particle aswell.We presented a viable choice for the spin dynamics sat-isfying both constraints [2] as will be also shown explicitlyin section II A˙ s µ = em F µν s ν + (cid:101) a (cid:18) F µν s ν − u µ c u · F · s (cid:19) − mc (cid:16) em + (cid:101) a (cid:17) s · ∂F ∗ µν s ν . (10)The first term ensures consistency with the Lorentzforce, second term introduces a magnetic anomaly forthe charged particles (cid:101) a satisfying dc = em + (cid:101) a (11)and the third term is necessary for consistency with theStern-Gerlach term in Eq. (5) through derivative of thecondition Eq. (8). We have chosen this form of writingthe constants because we can easily perform the limitof neutral particles, when charge e = 0 and the wholemagnetic moment is anomalous [22]. This allows us towrite for neutral particles a following set of equations˙ u µ = − s · ∂F ∗ µν u ν dm , (12)˙ s µ = cd (cid:18) F µν s ν − u µ c ( u · F · s ) (cid:19) − s · ∂F ∗ µν s ν dm . (13)These dynamical equations for neutral particles will be astarting point of our study. A. Justification: non-relativistic behavior
In the laboratory frame the velocity and spin and spin4-vectors read u µ = γc (1 , βββ ) , s µ = ( βββ · sss, sss ) , (14)where the spin 3 vector is a Lorentz transformation ofthe instantaneous co-moving frame spin sss c sss = sss c + γ − β ( βββ · sss c ) βββ . (15)The electromagnetic tensor and its dual in the laboratoryframe are F µν = (cid:18) − EEE/cEEE/c − (cid:15) ijk B k (cid:19) , F ∗ µν = (cid:18) BBB − BBB (cid:15) ijk E k /c (cid:19) . (16)Finally, the covariant gradient term can expressed as s · ∂ = βββ · sss ∂∂ct + sss · ∇ (17)The spatial part of the force equation Eq. (12) reads ddt ( γβββ ) = dm (cid:18) βββ · sss ∂∂ct + sss · ∇ (cid:19) ( BBBc − βββ × EEE ) , (18)and the spatial part of the torque equation Eq. (13) is γ ddtsss = d ( EEE ( βββ · sss ) + sss × BBBc ) − dγ βββ ( EEE · sss − ( βββ · EEE )( βββ · sss ) − βββ · ( sss × BBBc ) − dmc ( s · ∂ )( sss × EEE − BBBc ( βββ · sss )) (19)Following the derivation presented by Schwinger [23] wecan substitute Eq. (18) into Eq. (19) and in the non-relativistic limit we neglect terms quadratic in βββ andhigher˙ sss ≈ d ( EEE ( βββ · sss ) − βββ ( EEE · sss ) + sss × BBBc )+ ˙ βββ ( βββ · sss ) − dmc ( sss · ∇ )( sss × EEE ) (20)Substituting into the left hand side derivative of Eq. (15)and combining with the term containing ˙ βββ on the righthand side we obtain the Thomas precession term (cid:18) ddtsss c (cid:19) TP ≈ βββ × ˙ βββ × sss c , (21)which is independent of the size of the magnetic moment.In the instantaneous frame co-moving with the parti-cle we have βββ = 0, γ = 1 which further simplifies ourequations Eq. (18),19 to ddt ( vvv ) | c = = 1 m ( µµµ · ∇∇∇ ) BBB , (22) ddtsss | c = µµµ × (cid:18) BBB − mc ( sss · ∇∇∇ ) EEE c (cid:19) , (23) where we also rewrote spins in terms of the magnetic mo-ment using relationship Eq. (6). These expressions haveall the desired properties. The force equation Eq. (22)contains Gilbertian form of Stern-Gerlach interaction[1]. The torque equation Eq. (23) behaves according toconstraint in Eq. (9) - spin aligning with the magneticfield. In addition we have another term which dependson the component-wise gradient of the electric field inthe direction of the spin. This term is a new predic-tion of our theory, but a necessary addition in order tomake the torque equation Eq. (13) compatible with theStern-Gerlach force equation Eq. (12) through constraintEq. (8) which is not accounted for in the standard TBMTformulation [15]. A similar term, depending on quantummechanical model of spin, also arises from a classical limitof relativistic quantum equations, we will explore thiscorrespondence under a separate cover. Such additionalforce causes the spin vector not only to align with themagnetic field, but also depend on the gradient of theelectric field. B. Non-uniqueness of the torque equation
The proposed torque equation for the neutral particleEq. (23) is the simplest form which is consistent with theequation for the force Eq. (22) and generates the correcttorque term in the frame co-moving with the particle.We could imagine other terms, orthogonal to u µ whichcould be included in the torque equation. For examplewe can add˙ s µ = . . . + (cid:101) b (cid:18) s · ∂F ∗ µν s ν − u µ c u · ( s · ∂ ) F ∗ · s (cid:19) , (24)with (cid:101) b being another constant characterizing the classi-cal point particle. If we would repeat the analysis in thesection II A with this addition, the Thomas precessionterm Eq. (21) would be sensitive to the value of (cid:101) b . A pre-cession experiment with neutral particles could resolve ifsuch modification is necessary. III. SOLUTION FOR LINEARLY POLARIZEDPLANE WAVE
We consider potential of a planelinear polarized elec-tromagnetic wave A µ = ε µ A f ( ξ ) , ξ = ωc ˆ k · x , (25)where ε µ is polarization of the plane wave; ξ its invariantphase; A amplitude; and ˆ k µ a unit-less vector in thedirection of the wave vector. The wave vector is time-like and transverse to the space-like polarization vectorˆ k = 0 , ˆ k · ε = 0 , ε = − . (26) f ( ξ ) is a function characterizing the laser pulse contain-ing both the oscillatory part, and the pulse envelope.Using this 4-potential we can construct an electromag-netic field tensor F µν = ∂ µ A ν − ∂ ν A µ = A ωc f (cid:48) ( ξ )(ˆ k µ ε ν − ε µ ˆ k ν ) , (27)where prime denotes derivative with respect to the phase ξ . Notice that contraction of this tensor with ˆ k µ is zerobecause of properties Eq. (26). Another quantity of in-terest seen in Eq. (13) is( s · ∂ ) F ∗ µν = A ω c f (cid:48)(cid:48) ( ξ )(ˆ k · s ) (cid:15) µναβ ˆ k α ε β , (28)this time the contraction with both ˆ k µ or ε µ is zero be-cause of the anti-symmetry properties of (cid:15) µναβ .We rewrite the dynamical Eqs. (12, 13) for the planewave field using Eqs. (27, 28)˙ u µ = − A dω mc f (cid:48)(cid:48) ( ξ )(ˆ k · s ) (cid:15) µναβ u ν ˆ k α ε β , (29)˙ s µ = ωd A f (cid:48) ( ξ )(ˆ k µ ε · s − ε µ ˆ k · s ) − u µ ( u · F · s ) dc − A dω mc f (cid:48)(cid:48) ( ξ )(ˆ k · s ) (cid:15) µναβ s ν ˆ k α ε β . (30)In order to solve this system of equation we first look onthe dot product of these equations with ˆ k µ and ε µ whichallows us to find a differential equation for ˆ k · s ( τ ) (SectionIII A). Then we can solve for the particle dynamics u µ ( τ )(Section III B) and invariant acceleration (Section III C).Finally we will look on the motion in the laboratory frame(Section III D). A. Solutions for the spin projections
Contracting the first dynamical Eq. (29) with k µ weobtain a first integral of motionˆ k · ˙ u = ddτ (ˆ k · u ) = 0 , ⇒ ˆ k · u = ˆ k · u (0) . (31)This also allows us to find a relationship between thephase of the wave ξ and proper time of the particle dξdτ = ωc ddτ (ˆ k · x ) = ωc ˆ k · u (0) , ⇒ ξ = ωc (ˆ k · u (0)) τ + ξ . (32)Repeating the same line of argument with ε µ yields asecond integral of motion ε · ˙ u = ddτ ( ε · u ) = 0 , ⇒ ε · u = ε · u (0) . (33)Now we consider contractions of the second dynamicalEq. (30). Using the first integral of motion Eq. (31) thecontraction with k µ readsˆ k · ˙ s = − (ˆ k · u (0))( u · F · s ) dc . (34) Using now the second integral of motion Eq. (33) the con-traction with ε µ is ε · ˙ s = ωdA f (cid:48) ( ξ )(ˆ k · s ) − ( ε · u (0))( u · F · s ) dc . (35)The scalar quantity u · F · s can be evaluated using Eq. (27)and integrals of motion Eqs. (31, 33) u · F · s = A ωc f (cid:48) ( ξ ) W ( τ ) , (36)where W ( τ ) ≡ (ˆ k · u (0))( ε · s ( τ )) − ( ε · u (0))(ˆ k · s ( τ )) . (37)This allows us to write Eq. (34) asˆ k · ˙ s = −A ˙ f ( ξ ( τ )) W ( τ ) dc (38)where we absorbed the ω (ˆ k · u (0)) /c factor into the timederivative using differential of Eq. (32).We now take another proper time derivative of thisexpressionˆ k · ¨ s = −A ¨ f ( ξ ( τ )) W ( τ ) dc − A ˙ f ( ξ ( τ )) ˙ W ( τ ) dc (39)The term containing ˙ W ( τ ) can be simplified by pluggingboth projections Eq. (34) and Eq. (35) into the derivativeof the definition W ( τ ) Eq. (37) where the terms with ( u · F · s ) cancel leaving us with˙ W ( τ ) = cd A ˙ f ( ξ ( τ ))(ˆ k · s ) . (40)Next we note that the first term on the RHS of Eq. (39)can be expressed again in terms of (ˆ k · ˙ s ) using Eq. (38)leading us to the final dynamical equation for the spinprojectionˆ k · ¨ s = ¨ f ( ξ ( τ ))˙ f ( ξ ( τ )) (ˆ k · ˙ s ) − A d ˙ f ( ξ ( τ ))(ˆ k · s ) . (41)Equation (41) is a second order linear differential equa-tion for ˆ k · s ( τ ). The two general solutions areˆ k · s ( τ ) = exp ( ± i A df ( ξ ( τ ))) (42)as can be verified by a direct substitution. The truephysical solution can be found as their linear combinationsatisfying initial conditions which we will denote asˆ k · s ( τ = 0) ≡ ˆ k · s (0) , ˆ k · ˙ s (0) = −A ˙ f ( ξ ) W (0) dc , (43)where the initial condition for the derivative is given byEq. (38). After algebraic manipulations which heavily usetrigonometric identities we obtain as our final resultˆ k · s ( τ ) = ˆ k · s (0) cos [ A d ( f ( ξ ( τ )) − f ( ξ ))] − W (0) c sin [ A d ( f ( ξ ( τ )) − f ( ξ ))] . (44)It can be checked that this result satisfies the originaldynamical problem and the associated initial conditions.Note that if we consider a situation when the initial stateof the particle is long before the arrival of the plane wavepulse and final state long after it departed f ( ξ ( τ )) = f ( ξ ) and the projection ˆ k · s ( τ ) returns to its initialconfiguration ˆ k · s (0).Now we turn to solving for the projections ε · s ( τ ).Eliminating u · F · s from Eqs. (34, 35) yields ε · ˙ s = ωd A f (cid:48) ( ξ )(ˆ k · s ) + ε · u (0)ˆ k · u (0) ˆ k · ˙ s (45)and armed with the knowledge of solution for the ˆ k · s ( τ ) Eq. (44) we can integrate this equation imposing aninitial condition ε · s ( τ = 0) = ε · s (0) ε · s ( τ ) = ε · s (0) cos [ A d ( f ( ξ ( τ )) − f ( ξ ))]+ (cid:32) c ˆ k · s (0)ˆ k · u (0) − W (0) c ε · u (0)ˆ k · u (0) (cid:33) sin [ A d ( f ( ξ ( τ )) − f ( ξ ))] . (46)Again, long after the passing of the pulse the projection ε · s ( τ ) reinstates itself to the initial condition ε · s (0) longbefore the pulse’s arrival. B. Solution for the 4-velocity
Given the solutions for the projections of the 4-spin asa function of proper time we can solve for the particlemotion. We start with the first dynamical Eq. (29): wedivide by f (cid:48)(cid:48) ( ξ ( τ ))ˆ k · s ( τ ) and take another derivativewith respect to proper time ddτ (cid:32) ˙ u µ ( τ ) f (cid:48)(cid:48) ( ξ )ˆ k · s ( τ ) (cid:33) = − A dω mc (cid:15) µναβ ˙ u ν ( τ )ˆ k α ε β . (47)We can substitute for ˙ u ν on the right hand side backfrom the original dynamical equation Eq. (29) and con-tract the two anti-symmetric tensors while using integralsof motion Eqs. (31, 33) as follows (cid:15) µναβ (cid:15) νργδ ˆ k α ε β u ρ ( τ )ˆ k γ ε δ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ µρ δ µγ δ µδ δ αρ δ αγ δ αδ δ βρ δ βγ δ βδ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ k α ε β u ρ ( τ )ˆ k γ ε δ == (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u µ ( τ ) ˆ k µ ε µ ˆ k · u (0) 0 0 ε · u (0) 0 − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (ˆ k · u (0))ˆ k µ . (48)This gives us an expression ddτ (cid:32) ˙ u µ ( τ ) f (cid:48)(cid:48) ( ξ )ˆ k · s ( τ ) (cid:33) = A d ω m c (ˆ k · s ( τ )) f (cid:48)(cid:48) ( ξ )(ˆ k · u (0))ˆ k µ , (49)which can be formally integrated by introducing a unit-less function h ( τ ) ≡ − A dω mc (cid:90) ττ =0 ˆ k · s ( (cid:101) τ ) f (cid:48)(cid:48) ( ξ ( (cid:101) τ )) d (cid:101) τ , (50) this integral has to be computed for a specific laser pulsefield. Let’s integrate equation Eq. (49) twice using initialconditions u µ ( τ = 0) ≡ u µ (0) , (51)˙ u µ ( τ = 0) = − A dω mc f (cid:48)(cid:48) ( ξ )(ˆ k · s (0)) (cid:15) µναβ u ν (0)ˆ k α ε β , (52)where the initial condition for the derivative is given byEq. (29). We obtain the final solution u µ ( τ ) = u µ (0) + 12 h ( τ )ˆ k µ (ˆ k · u (0))++ h ( τ ) (cid:15) µναβ u ν (0)ˆ k α ε β . (53)Finally, we also want to evaluate the expression (cid:15) µναβ u ν ˆ k α (cid:15) β = (cid:15) µναβ u ν (0)ˆ k α (cid:15) β + h ( τ )(ˆ k · u (0))ˆ k µ , (54)where we used the solution Eq. (53) and contraction iden-tity for the anti-symmetric tensors. This equation willprove to be useful in the Section III D when we evalu-ate the motion in the laboratory frame as it capturesthe motion in the plane perpendicular to the wave vectorand the polarization direction. Three 4-vectors k µ , ε µ ,and (cid:15) µναβ u ν (0) k α ε β are mutually 4D-orthogonal and canbe taken as a covariant basis of a 3D-subspace of theMinkowski space. C. Invariant acceleration of the particle
Another quantity of interest is the invariant acceler-ation which can be obtained by squaring Eq. (29) andcontracting the anti-symmetric tensors˙ u ( τ ) = − (cid:18) d A ω mc (cid:19) f (cid:48)(cid:48) ( ξ ) (ˆ k · s ( τ )) (ˆ k · u (0)) . (55)This expression is completely defined by the secondderivative of the pulse function f (cid:48)(cid:48) ( ξ ) and by the derivedsolution for ˆ k · s ( τ ) Eq. (44). D. Laboratory frame quantities
In the laboratory (lab) frame the relevant initial 4-vectors read u µ (0) = γ c (1 , βββ ) , ˆ k µ = (1 , ˆ kkk ) , ε µ = (0 , εεε ) . (56)The initial spin 4-vector can be obtained by imposing acondition u (0) · s (0) = 0 and Lorentz transformation ofthe instantaneous co-moving frame spin (0 , sss ) s µ (0) = ( βββ · sss L , sss L ) , sss L = sss + γ − β ( βββ · sss ) βββ . (57) FIG. 1. Coordinates in the laboratory frame are chosenso that unit vector εεε points in the direction of x -axis, ˆ kkk × εεε in the direction of y -axis, and ˆ kkk in the direction of z -axis. α , α , and α are direction cosines with respect to these axesrespectively. All initial conditions can be expressed in terms of thelaboratory frame quantities as followsˆ k · u (0) = γ c (1 − ˆ kkk · βββ ) , (58) ε · u (0) = − γ cεεε · βββ , (59) ε · s (0) = − εεε · sss − ( γ − βββ · sss )( εεε · ˆ βββ ) , (60)ˆ k · s (0) = γ βββ · sss − ˆ kkk · sss − ( γ − βββ · sss )(ˆ kkk · ˆ βββ ) . (61)The first integral of motion Eq. (31) can be expressed inthe lab frame as γ (1 − ˆ kkk · βββ ) = γ (1 − ˆ kkk · βββ ) , (62)and similarly for the second integral of motion Eq. (33) γεεε · βββ = γ εεε · β β β . (63)Let’s define γ ( τ ) ≡ γ (1 + G ( τ )) , (64)where G ( τ ) is a measure of how big is the difference be-tween the instantaneous γ -factor γ ( τ ) and initial γ -factor γ relative to γ . Now the zeroth component of the4-velocity solution Eq. (53) tells us how is the γ -factorchanging as a function of the particle’s proper time. Interms of G ( τ ) we have G ( τ ) ≡ h ( τ )(1 − ˆ kkk · βββ ) + h ( τ ) βββ · (ˆ kkk × εεε ) . (65)This expression also allows us to find the magnitude ofvelocity as a function of proper time β ( τ ) = 1 − − β (1 + G ( τ )) . (66) Taking a zeroth component of Eq. Eq. (54) gives us γβββ · (ˆ kkk × εεε ) = γ βββ · (ˆ kkk × εεε ) + γ h ( τ )(1 − ˆ kkk · βββ ) ωc . (67)We introduce three directional cosines which are pro-jections of the βββ vector on each of the coordinate axes indirection of unit vectors, εεε , ˆ kkk × εεε , and ˆ kkk shown in Fig. 1 εεε · βββ = β cos α , (ˆ kkk × εεε ) · βββ = β cos α , ˆ kkk · βββ = β cos α . (68)Using expressions for γ ( τ ) Eq. (64) and β ( τ ) Eq. (66) inthe second integral of motion Eq. (63) determines howthe first directional cosine α ( τ ) changescos α ( τ ) = β cos α (0) (cid:112) β + G ( τ ) + 2 G ( τ ) . (69)Similarly, substituting into Eq. (67) gives us an expres-sion for α ( τ )cos α ( τ ) = β cos α (0) + h ( τ )(1 − β cos α (0)) ω/c (cid:112) β + G ( τ ) + 2 G ( τ ) . (70)And finally, the third directional cosine α ( τ ) can be ob-tained by substituting all the quantities into first integralof motion Eq. (62)cos α ( τ ) = G ( τ ) + β cos α (0) (cid:112) β + G ( τ ) + 2 G ( τ ) . (71)Now we can easily switch to the usual spherical angles θ ( τ ) , φ ( τ ) using formulascos θ ( τ ) = cos α ( τ ) , tan φ ( τ ) = cos α ( τ )cos α ( τ ) , (72)leading us tocos θ ( τ ) = G ( τ ) + β cos θ (cid:112) β + G ( τ ) + 2 G ( τ ) , (73)andtan φ ( τ ) = tan φ + h ( τ ) ωc (cid:20) − β cos θ β sin θ cos φ (cid:21) . (74)The meaning of the conserved quantities k · u (0) and ε · u (0) or in the laboratory frame the equations Eq. (62)and Eq. (63) is that particle can lower its velocity whileincreasing the angle θ of its direction of motion with re-spect to ˆ kkk and vice versa. It also means that the geome-try dictates a minimal velocity for the particle given theinitial conditions. From Eqs. (62,64,68) G ( τ ) > β sin θ β cos θ − . (75) FIG. 2. Cosine of the angle between βββ and ˆ kkk (EquationEq. (73)) plotted as a function of G ( τ ) for different initialvalues cos θ ∈ [ − ,
1] which correspond to G ( τ ) = 0. Theultra-relativistic limit β ≈ IV. ULTRA-RELATIVISTIC NEUTRINOS
The primary objective of this study is to show if Diracneutrinos can be deflected in their path by intense laserfields. This would mean that the neutrino beams that arefocused on detectors far away for the purpose of study ofneutrino oscillations would experience variation in theevent count as a function of applied pulsed laser field,where neutrino pulses and laser pulses are synchronized.We will estimate the magnitude of the effect as a functionof both laser and neutrino properties.We rewrite the amplitude of the laser field A usingthe dimensionless normalized amplitude a A = m e ce a . (76)The conversion between the magnetic moment and ele-mentary dipole charge of particle d reads d = emc µ ν [ µ B ] , (77)where µ ν is in units of Bohr magnetons. This makes theproduct A d appearing in our equations A d = a µ ν [ µ B ] ≈ − − − , (78)for the state of the art laser systems with a ∼ andwhole range of possible magnetic moments of the Diracneutrinos Eqs. (1,2). Such laser system is currently underconstruction in ELI Beamlines, Prague and the dimen-sionless normalized amplitude for this laser correspondsto the power of 10 PW [24] with an intensity 10 W/cm .Since the product A d is so small the arguments of thetrigonometric functions in the solutions Eqs. (44, 46) arenegligible and the solutions reduce toˆ k · s ( τ ) ≈ ˆ k · s (0) , ε · s ( τ ) ≈ ε · s (0) . (79) In other words there is no precession in these directions.For ultra-relativistic neutrinos β ≈ G ( τ ) >
0, the neutrinos have tendency tofocus - cosine of the angle between the velocity of neutri-nos and wave vector of the laser light Eq. (73) approachesone as γ ( τ ) increases.As we showed in our previous work [2] the square rootof the invariant acceleration Eq. (55) is greatly increasedby the initial gamma factor of the neutrino E ν /m ν squared. Now that we have a solution for the particlemotion we can estimate how the gamma factor Eq. (64)and the angle Eq. (73) of the neutrino are changing withthe proper time. A. Estimate of change for velocity and angle
Both the angle Eq. (73) and the gamma factor Eq. (64)depend only on the value of the function G ( τ ). Thisfunction Eq. (65) can be evaluated by explicitly integrat-ing h ( τ ) Eq. (82) with a specific laser pulse oscillationsand profile. In the ultra-relativistic limit Eq. (79) thisintegral can be estimated as h ( τ ) ≈ − A dωmc ˆ k · s (0)ˆ k · u (0) f (cid:48) ( ξ ( τ )) , (80)where we assumed that we started counting the propertime long before the pulse arrived when f (cid:48) ( ξ ) = 0. Fromthe lab frame quantities Eqs. (58, 61) the fractionˆ k · s (0)ˆ k · u (0) ≈ βββ · sss c ≈ (cid:126) c , (81)which does not depend on the initial gamma factor. Us-ing the estimate Eq. (78) the expression for h ( τ ) is h ( τ ) ≈ − a µ ν [ µ B ] E γ [ eV ] m ν [ eV ] f (cid:48) ( ξ ( τ )) ≈ − (10 − − − ) f (cid:48) ( ξ ( τ )) , (82)where E γ is the energy of the laser photons and the nu-merical value was calculated for E γ = 1 eV because ELIBeamlines will operate in the visible range. Our resultEq. (82) shows that the effect depends on the product of a with E γ which is proportional to the square root oflaser intensity. There are lasers with much higher elec-tron energy (like free electron lasers XFEL in Hamburg[25] or LCLS-II under construction in Stanford [26]), buttheir intensity is lower for coherent photons from a givenenergy band. Moreover, lasers with higher photon en-ergy have shorter wavelength which would invalidate ourcondition of classical limit Eq. (3) for some sources (Forexample neutrinos from beta decay have energy ∼ λ ν ∼ − m and 10 keV photonhas wavelength ∼ − m). Mass of neutrino was takenas m ν = 0 . γ . Looking at the equation for G ( τ ) - equationEq. (65) the square of h ( τ ) is completely negligible sothat if we keep only the linear term in h ( τ ) we get G ( τ ) ≈ (10 − − − ) f (cid:48) ( ξ ( τ )) , (83)which means that unless we can prepare laser with veryhigh derivative f (cid:48) ( ξ ) in which our approximation Eq. (79)is no longer valid, the changes of the gamma factor andangle will be minuscule.Note that this ultra-relativistic limit is also not validfor neutrons because the product A d is not negligibleanymore and the projection k · s ( τ ) is no longer constant- the spin precesses - and has to be taken into accountin the integral 82. We wish to return to the neutrondynamics in the laser field in the future. V. DISCUSSION AND CONCLUSIONS
In this paper we formulated and explored a covariantclassical neutral particle dynamics in external EM fields.The neutral particle interacts with the fields through itsmagnetic moment and in the co-moving frame feels aStern-Gerlach force Eq. (22) and torque Eq. (23). Ourcovariant formulation clarifies that apart from the usualtorque term µµµ × BBB we also have a term proportional to thegradient of the electric field in the direction of the spinwhich is necessary to satisfy the constraint of spin and4-velocity orthogonality Eq. (8) while keeping the Stern-Gerlach force intact.In the proposed formulation we restricted ourselves to the natural form of the spin dynamics, however we can-not exclude that other terms orthogonal to u µ can beadded to the dynamical equation for spin. Thereforein the section II B we discuss the non-uniqueness of thetorque equation and possibility of adding another termwhich would change the Thomas precession coefficient.While we continue our search for theoretical rationalethat would uniquely define the torque equation, we notethat a precession experiment with neutral particles withnonzero magnetic moment can determine if such modifi-cation is present.We presented an analytical solution of our dynamicalequations in the external linearly polarized electromag-netic plane wave field. We showed that the projections ofparticle 4-velocity on the wave and polarization 4-vectorsare constants of the motion Eqs. (31,33). We formulateda differential equation for the projection of the 4-spin k · s ( τ ) and found its solution Eq. (44). Finally, we solvedfor the 4-velocity of the particle Eq. (53).Our results are obtained in the classical dynamicsframework. Several decades ago Skobelev [20] consideredquantum field theory formulation of processes in the pres-ence of the magnetic / electrical dipole of neutrino. Forthe state of the art fields of that time period the effect wasnot measurable. However, if we extrapolate the progressin laser technology made since this work appeared, we re-main optimistic that experiments studying this effect willbecome possible. While quantum approach may provideadditional motivation for selection of classical dynamicalequations, considering the short de Broglie wavelengthof ultra-relativistic particles, classical dynamics may suffhow these results can be used in neutron and neutrinobeam control, allowing in the case of neutrinos to obtaininformation about their properties. [1] J. Rafelski, M. Formanek, and A. Steinmetz, “Relativis-tic dynamics of point magnetic moment,” The EuropeanPhysical Journal C , vol. 78, no. 1, p. 6, 2018.[2] M. Formanek, S. Evans, J. Rafelski, A. Steinmetz, andC.-T. Yang, “Strong fields and neutral particle magneticmoment dynamics,”
Plasma Physics and Controlled Fu-sion , vol. 60, no. 7, p. 074006, 2018.[3] E. Sarachik and G. Schappert, “Classical theory of thescattering of intense laser radiation by free electrons,”
Physical Review D , vol. 1, no. 10, p. 2738, 1970.[4] C. Itzykson and J.-B. Zuber,
Quantum field theory .Courier corporation, 2005.[5] J. Rafelski, “Electrons riding a plane wave,” in
RelativityMatters , pp. 343–358, Springer, 2017.[6] Y. Hadad, L. Labun, J. Rafelski, N. Elkina, C. Klier,and H. Ruhl, “Effects of radiation reaction in relativisticlaser acceleration,”
Physical Review D , vol. 82, no. 9,p. 096012, 2010.[7] A. Di Piazza, “Exact solution of the landau-lifshitz equa-tion in a plane wave,”
Letters in Mathematical Physics ,vol. 83, no. 3, pp. 305–313, 2008.[8] R. N. Mohapatra, S. Antusch, K. Babu, G. Barenboim, M.-C. Chen, A. De Gouvˆea, P. De Holanda, B. Dutta,Y. Grossman, A. Joshipura, et al. , “Theory of neutrinos:a white paper,”
Reports on Progress in Physics , vol. 70,no. 11, p. 1757, 2007.[9] T. Kotani, E. Takasugi, et al. , “Double beta decay andmajorana neutrino,”
Progress of Theoretical Physics Sup-plement , vol. 83, pp. 1–175, 1985.[10] R. Henning, “Current status of neutrinoless double-betadecay searches,”
Reviews in Physics , vol. 1, pp. 29–35,2016.[11] K. Fujikawa and R. E. Shrock, “Magnetic moment of amassive neutrino and neutrino-spin rotation,”
PhysicalReview Letters , vol. 45, no. 12, p. 963, 1980.[12] C. Patrignani, P. D. Group, et al. , “Review of particlephysics,”
Chinese physics C , vol. 40, no. 10, p. 100001,2016.[13] L. H. Thomas, “The motion of the spinning electron,”
Nature , vol. 117, no. 2945, p. 514, 1926.[14] J. Frenkel, “Die elektrodynamik des rotierenden elek-trons,”
Zeitschrift f¨ur Physik , vol. 37, no. 4-5, pp. 243–262, 1926.[15] V. Bargmann, L. Michel, and V. L. Telegdi, “Precession of the polarization of particles moving in a homogeneouselectromagnetic field,”
Physical Review Letters , vol. 2,no. 10, p. 435, 1959.[16] A. J. Silenko, “Foldy-wouthyusen transformation andsemiclassical limit for relativistic particles in strong ex-ternal fields,”
Physical Review A , vol. 77, no. 1, p. 012116,2008.[17] A. Steinmetz, M. Formanek, and J. Rafelski, “Magneticdipole moment in relativistic quantum mechanics,”
TheEuropean Physical Journal A , vol. 55, no. 3, p. 40, 2019.[18] M. Wen, H. Bauke, and C. H. Keitel, “Identifying thestern-gerlach force of classical electron dynamics,”
Sci-entific reports , vol. 6, p. 31624, 2016.[19] M. Wen, C. H. Keitel, and H. Bauke, “Spin-one-half par-ticles in strong electromagnetic fields: Spin effects andradiation reaction,”
Physical Review A , vol. 95, no. 4,p. 042102, 2017.[20] V. Skobelev, “Interaction between a massive neutrinoand a plane wave field,”
Sov. Phys. JETP , vol. 73, p. 40,1991.[21] P. D. Morley and D. J. Buettner, “Instantaneous powerradiated from magnetic dipole moments,”
Astroparticle Physics , vol. 62, pp. 7–11, 2015.[22] J. J. Sakurai,
Advanced quantum mechanics . PearsonEducation India, 1967.[23] J. Schwinger, “Spin precessiona dynamical discussion,”
American Journal of Physics , vol. 42, no. 6, pp. 510–513,1974.[24] B. Rus, P. Bakule, D. Kramer, J. Naylon, J. Thoma,J. Green, R. Antipenkov, M. Fibrich, J. Nov´ak,F. Batysta, et al. , “Eli-beamlines: development of nextgeneration short-pulse laser systems,” in
Research UsingExtreme Light: Entering New Frontiers with Petawatt-Class Lasers II , vol. 9515, p. 95150F, International Soci-ety for Optics and Photonics, 2015.[25] M. Altarelli, R. Brinkmann, and M. Chergui, “The eu-ropean x-ray free-electron laser. technical design report,”tech. rep., DEsY XFEL Project Group, 2007.[26] J. Galayda, “The linac coherent light source-ii project,”in