Induced Lorentz Violation on a Moving Braneworld

We consider a braneworld scenario in which a flat 4-D brane, embedded in $M^{3,1} \times S^1$, is moving on or spiraling around the $S^1$. Although the induced metric on the brane is 4-D Minkowski, the would-be Lorentz symmetry of the brane is broken globally by the compactification. As recently pointed out this means causal bulk signals can propagate superluminally and even backwards in time according to brane observers. Here we consider the effective action on the brane induced by loops of bulk fields. We consider a variety of self-energy and vertex corrections due to bulk scalars and gravitons and show that bulk loops with non-zero winding generate UV-finite Lorentz-violating terms in the 4-D effective action. The results can be accommodated by the Standard Model Extension, a general framework for Lorentz-violating effective field theory.


Introduction
The simplest braneworld scenario posits a spacetime of the form M 3,1 × S 1 , with a single extra dimension compactified on a circle of radius R. The brane, assumed to be at a fixed position on the S 1 , has a Minkowski metric induced on its worldvolume.In this scenario worldvolume Lorentz invariance is an exact symmetry, inherited from a symmetry of the underlying 5-D spacetime.
A straightforward generalization of this scenario allows the brane to either move on or spiral around the S 1 .This generalization might seem quite innocuous.The induced metric on the brane is still 4-D Minkowski, so it would seem that brane observers might be hard-pressed to find any evidence that their brane has been boosted or rotated into the compact direction.
From a different perspective, however, the effects of this generalization are quite dramatic.Compactification on S 1 preserves an SO(3, 1) symmetry that acts on the directions orthogonal to the S 1 .Once the brane is moving on or spiraling around the S 1 this exact SO (3,1) symmetry no longer aligns with the would-be Lorentz symmetry of the brane worldvolume.Although there is no local indication of the violation, worldvolume Lorentz invariance is broken globally by the compactification.Without Lorentz invariance all bets are off, and indeed [1,2,3] showed that bulk signals can propagate faster than light and even backwards in time according to brane observers.Fortunately causality -a more robust feature -remains intact, inherited from the causality of the underlying 5-D spacetime.
Here we consider the effects of virtual bulk particles in this generalized braneworld scenario.Such particles can leave a moving or rotated brane, propagate around the compact dimension, and return.Bulk loops have no reason to respect worldvolume Lorentz symmetry and might be expected to induce Lorentz-violating terms in the brane effective action.We will see that this is indeed the case.We focus on Lorentz-violating dimension-4 terms in the effective action, especially the electron self-energy and the electron -photon vertex, and show that bulk loops induce specific Lorentz-violating terms with finite, calculable coefficients.These terms are part of the Standard Model Extension, a general framework for Lorentz-violating effective field theories developed in [4,5].There are stringent experimental bounds on the Lorentz-violating coefficients which have been tabulated in [6].
An outline of this paper is as follows.In section 2 we describe the braneworld scenario we will consider.Section 3 discusses the propagator for a bulk scalar field.In sections 4 and 5 we evaluate corrections to the electron self-energy and the electron -photon vertex due to a bulk scalar loop.Section 6 considers the self-energy for a scalar field on the brane induced by a bulk scalar.Sections 7 and 8 evaluate the electron and scalar self-energy induced by a bulk graviton loop.We conclude in section 9 with a discussion of experimental bounds and future directions.
Cosmological implications of this scenario have been studied in [7] and a different approach to braneworld Lorentz violation has been developed in [8].

Boosted and rotated branes
Consider a 5-D spacetime M 3,1 × S 1 .To describe this we begin from 5-D Minkowski space M 4,1 with coordinates and metric We obtain an S 1 by periodically identifying the Z coordinate, Z ≈ Z + 2πR.It's convenient to describe this identification as These upper-case coordinates define the preferred frame for the compactification, with an exact SO(3, 1) symmetry that acts on the coordinates X µ .
The standard braneworld scenario would be to place a 4-D braneworld at rest at Z = 0. We are instead interested in braneworld which is moving in the Z direction and / or has been rotated into the Z direction.To describe this we transform to a new frame with lower-case coordinates x M via Here L M N is an SO(4, 1) transformation that acts non-trivially on the Z coordinate.In the x M coordinates there is a boosted and / or rotated identification We set and imagine a braneworld at z = 0.The coordinates x M can be thought of as co-moving and / or co-rotated with the brane.Since all we've done is a 5-D Lorentz transformation, in the co-moving coordinates the metric still has the form The compactification is hidden in the identification (4).Thus the induced metric on the brane is 4-D Minkowski, however the SO(3, 1) symmetry of the brane metric does not align with the SO(3, 1) symmetry that is preserved by the compactification (2).Instead worldvolume Lorentz symmetry is broken globally by the compactification, which leads to the curious possibilities of superluminal and even backwards-in-time signaling explored in [1,2,3].
From the brane point of view it's natural to decompose a M into components tangent and normal to the brane, so we set a µ becomes a preferred 4-vector on the brane, which shows that 4-D Lorentz symmetry on the brane is spontaneously broken.The fifth component 2πr is a scalar on the brane.
In the calculations below we will find it useful to work with the combination Up to Lorentz transformations on the brane there are three cases to consider.

Timelike b µ
In this case we can go to a reference frame on the brane in which b µ only has a time component.This can be obtained directly from (2) by boosting with velocity β in the Z direction.
This leads to or alternatively This corresponds to the "boost-like isotropic" case discussed in [3].As seen on the brane, bulk signals propagate isotropically in all directions at superluminal speeds.

Spacelike b µ
In this case we can go to a reference frame on the brane in which b µ only has an x component.This can be obtained directly from (2) by rotating through an angle θ in the XZ plane.
This leads to a µ = (0, sin θ 2πR, 0, 0) Note that b µ = (0, tan θ, 0, 0) ( 16) or alternatively This corresponds to the "tilt-like anisotropic" case discussed in [3].As seen on the brane, bulk signals propagate superluminally in the x direction and at the speed of light in perpendicular directions.

Lightlike b µ
Finally we consider the case of lightlike b µ . 1 This can be obtained starting from (2) by making a Lorentz rotation in the X − Z plane. 2 Here we've introduced light-front coordinates X ± = T ± X with The form of the Lorentz transformation is a little unfamiliar.Introducing a parameter λ ∈ R it takes the form To see that this is the appropriate Lorentz transformation note that it leaves X + invariant, X + = x + , so it acts on X − Z planes.Also it preserves the metric (19), with Applying the (inverse of) the transformation (20) gives So the radius is unchanged, while is indeed a null vector on the brane.
A null vector has no invariant length, so one can go to an infinitely-boosted frame in which b µ = 0.This restores conventional Lorentz invariance on the brane.However if any matter (e.g.CMB photons) is present on the brane one may not wish to perform an infinite boost.In section 3 we show that when λ is non-zero bulk signals can have a negative light-front velocity in the x − direction.With respect to Minkowski time this means that as seen on the brane a bulk signal can travel faster than light and even backwards in time in the x direction.For further discussion of the geometry of this case see appendix A.
Note that in all three cases we have b 2 < 1.The range −∞ < b 2 < 0 is tilt-like, b 2 = 0 is null, and 0 < b 2 < 1 is boost-like.Alternatively we can say that we always have 1

Bulk scalar propagator
We expect that bulk loops should induce Lorentz-violating terms on the brane.Before turning to explicit calculations we start with a discussion of the bulk propagator.We focus on bulk scalar fields for simplicity.
The retarded propagator for a bulk field was discussed in [1] while the static Green's function was studied in [9].Here we consider the Feynman propagator.It's straightforward to impose the appropriate periodicity (x µ , z) ≈ (x µ + a µ , z + 2πr) using a winding sum (equivalently, a sum over image charges).In position space this gives the propagator for a bulk scalar of mass µ as We set z = 0 since we will only be interested in bulk propagation that starts and ends on the brane.Also we work in momentum space along the brane, which amounts to dropping x .Then we are left with the winding-sum form for the bulk propagator, We can switch from a sum over windings to a sum over Kaluza-Klein momentum using the Poisson resummation identity This puts the bulk propagator in the form It's clear that b µ ̸ = 0 breaks 4-D Lorentz invariance.We can look for poles in the propagator and read off the dispersion relation for the Kaluza-Klein tower, to see how it is modified from the perspective of a moving or rotated brane [9,3].There are three cases to consider.

Timelike b µ
In this case we set b µ = (−β, 0, 0, 0) and k µ = (ω, k).The propagator has poles at One branch of solutions has ω > 0 and a pole that is displaced slightly below the real axis.The other branch has ω < 0 and a pole that is displaced slightly above the real axis.Although we don't have 4-D Lorentz invariance, the poles are displaced in the standard way that allows for a Wick rotation to Euclidean signature.One can check that there are no tachyons from a 4-D perspective, ω 2 − |k| 2 ≥ 0. Finally we can evaluate the group velocity This makes it clear that wave propagation is isotropic, with a velocity 0 ≤ v g < γ that exceeds the speed of light if |k| is sufficiently large.

Spacelike b µ
In this case we set b µ = (0, tan θ, 0, 0) and k µ = (ω, k x , k ⊥ ).The propagator has poles at Again one branch of solutions has ω > 0 and a pole that is displaced slightly below the real axis, while the other branch has ω < 0 and a pole that is displaced slightly above the real axis, so we can perform Wick rotation in the standard way.One can check that there are no tachyons from a 4-D perspective, Finally the group velocity is anisotropic.For a wave propagating in the x direction while for a wave propagating in one of the perpendicular directions In the perpendicular directions we have the familiar group velocity for a Kaluza-Klein tower of particles with masses µ 2 n = n r 2 + µ 2 .In the x direction we have a group velocity 0 ≤ |v gx | < 1 cos θ that exceeds the speed of light if |k x | is sufficiently large.
We'll interpret τ = x + as light-front time and the conjugate momentum k + = 1 2 k − as light-front energy.The propagator has poles at This fixes the dispersion relation.As usual there are two branches of solutions.
Positive frequency modes have k + > 0 and k − > 0, while negative frequency modes have k + < 0 and k − < 0. Given a positive-frequency plane-wave solution a stationary-phase approximation lets us read off the group velocities with respect to light-front time.
In the transverse directions we have conventional light-front kinematics. 4In the longitudinal direction there is a shift which allows the longitudinal velocity to be negative, −λ 2 < v − < ∞.This means that in Minkowski coordinates bulk signals can travel faster than light and even backwards in time in the x direction.To see this note that in Minkowski coordinates a trajectory x − = −λ 2 x + corresponds to The Minkowski velocity is superluminal for 0 < λ 2 < 1.The velocity diverges at λ 2 = 1 and becomes negative for λ 2 > 1, which as in [2] indicates that the signal is traveling backwards in time.For further discussion of this case see appendix A.

Electron self-energy
The world-volume metric induced on the brane is 4-D Minkowski, even if the brane is boosted or rotated in the Z direction.Particles that solely propagate on the brane are not sensitive to the breaking of 4-D Lorentz invariance and it would be reasonable to describe these "standard model" particles using an effective action with 4-D Lorentz symmetry.However particles that propagate in the bulk can leave the brane, travel around the compactification manifold, and return.Such particles notice the global breaking of 4-D Lorentz invariance by the compactification and loops of such particles should induce Lorentz-violating terms in the 4-D effective action.
Here we study this effect, beginning with the simple example of radiative corrections to the electron self-energy.We imagine a real bulk scalar field χ of mass µ that has a Yukawa coupling to the electron.We describe the coupled system with the action Note that the coupling λ has units (mass) −1/2 .The diagram we wish to consider is shown in Fig. 1.
Our goal is to evaluate the diagram and expand in powers of the external momentum p.In this way we will make contact with the Standard Model Extension, a general effective field theory framework for Lorentz-violating effects developed in [4,5].The basic diagram is easy to write down.Writing the brane-to-brane bulk propagator with a sum over Kaluza-Klein momentum as in (29) we have As pointed out in section 3, even though the bulk propagator is not Lorentz invariant, it still has poles that allow for a standard Wick rotation.So we Wick rotate in the usual way, setting with a similar rotation for all other 4-vectors.We introduce a pair of Schwinger parameters s, t to represent the propagators via the identity It's convenient to use a frame in which only the first component of b E is non-zero.
The momentum integrals are Gaussian and lead to the rather tedious expression Now we expand in powers of the external momentum p.At zeroth order, after continuing back to Lorentzian signature and restoring Lorentz covariance, we find The term proportional to m is Lorentz invariant and therefore not interesting to us.The term proportional to / b has the potential to violate Lorentz invariance, but it vanishes once the sum over n is performed.This follows from a symmetry: the underlying expression (47) is invariant under b E → −b E , n → −n which implies that only even powers of b µ can appear.
At first order in p µ , after continuing back to Lorentzian signature and restoring Lorentz covariance, we find Here we've used the identity (28) in reverse to replace the momentum sum with a winding sum and an integral over q.The integral over q is Gaussian and leads to (49).
Working with a winding sum is advantageous for the following reasons.
• Lorentz symmetry is broken globally by the compactification.Particle trajectories with w = 0 are not sensitive to the breaking and are guaranteed to respect Lorentz invariance.Indeed in (49) we see that the term with w = 0 is proportional to / p.
• Ultraviolet divergences can only arise from the w = 0 term in the sum, since nonzero winding means the loop can never shrink to a point.Indeed in (49) we see that for w ̸ = 0 the exponential in the second line serves to cut off the short-distance regime s, t → 0.
Since we are only interested in Lorentz-violating terms, we could simply discard the w = 0 term to obtain a finite result.However we might as well discard all terms proportional to / p.This means discarding the first term in parenthesis in (49) as well as the trace part of b µ b ν .In this way we obtain the Lorentz-violating contribution This corresponds to a Lorentz-violating term in the 4-D effective Lagrangian for the electron.In the notation of [5] the relevant term is L = ic µν ψγ µ ∂ ν ψ which makes a contribution ic µν γ µ p ν to iΣ.Comparing to (50) we can read off the Lorentz violating coefficient c µν , which can be conveniently presented as where we've defined5 (52) The induced coefficients c µν are real, dimensionless, traceless and symmetric.They make a Lorentz-violating but CPT-even contribution to the effective action.
We can think of (51) as a product of a loop factor 1 16π 2 , a dimensionless coupling λ 2 πr , a tensor structure b µ b ν − 1 4 η µν b 2 , and a function I 1 of the dimensionless parameters b 2 , πmr, πµr.As can be seen in Fig. 2, I 1 is an increasing function of b 2 .It vanishes as b 2 → −∞ and (perhaps despite appearances) approaches a finite limit as b 2 → 1.
The expression for I 1 simplifies if we set m = 0 (a massless fermion on the brane) and b 2 = 0 (a small boost and / or rotation).Then the sum and integrals can be performed and the behavior for small and large µr can be extracted. 6This leads to

Electron -photon vertex
Next we consider the one-loop correction to the electron -photon vertex due to a bulk scalar.The diagram is shown in Fig. 3.
Suppressing the external polarizations and writing the bulk propagator with a momentum sum as in (29), the diagram is We can evaluate this similarly to the electron self-energy.We Wick rotate, introduce a series of Schwinger parameters , and evaluate the Gaussian integral over k µ E .This gives where we've introduced the convenient notation Now we expand in powers of the external momenta.At leading (zeroth) order, after continuing back to Lorentzian signature, switching to a winding sum for the bulk propagator and doing a bit of Dirac algebra, we find We drop all Lorentz-invariant terms, which includes the UV-divergent terms with w = 0. Setting s = s 1 + s 2 we're left with the Lorentz-violating contribution This pairs nicely with (50) to produce a gauge-invariant but Lorentz-violating dimension-4 term in the effective action, namely The coefficient c µν is given in (51).Since we stopped at zeroth order in the momentum this outcome, required by gauge invariance and Ward identities, can be thought of as a consistency check on our results.Expanding (56) beyond zeroth order in the external momenta would give higher-derivative corrections to the electron -photon vertex.

Scalar self-energy
Having calculated the one-loop Lorentz-violating correction to the self-energy of an electron, we now perform a similar calculation for a real scalar field ϕ on the brane with a cubic coupling to a bulk scalar χ.We start from the action Note that the coupling g has units (mass) +1/2 .The diagram we wish to consider is shown in Fig. 4.
Writing the brane-to-brane bulk propagator with a sum over Kaluza-Klein momentum as in (29), the diagram is We Wick rotate to Euclidean signature as in (43) and introduce a pair of Schwinger parameters as in (44).Parametrizing the Euclidean momenta as in (45) and performing the Gaussian integral over k E we find Now we expand in powers of the external momentum p.At zeroth order the result is Lorentz invariant and can be ignored.At first order the sum over Kaluza-Klein momentum vanishes because it is odd under n → −n.At second order, after continuing back to Lorentzian signature and restoring Lorentz covariance, we find Again we've used the identity (28) in reverse to replace the momentum sum with a winding sum and an integral over q.The integral over q is Gaussian and leads to (64).The first term in parenthesis is Lorentz invariant and can be dropped.The second term can be matched to a Lorentz-violating term in the effective action [5] with a traceless coefficient k µν .Removing the Lorentz-invariant trace from the second term in (64) we identify where I n is defined in (52).The induced coefficients k µν are real, dimensionless, traceless and symmetric.They make a Lorentz-violating but CPT-even contribution to the effective action.
We can think of (66) as a product of a loop factor 1 16π 2 , a dimensionless coupling g 2 πr, a tensor structure b µ b ν − 1 4 η µν b 2 , and a function I 2 of the dimensionless parameters b 2 , πmr, πµr.As can be seen in Fig. 5, I 2 is an increasing function of b 2 .It vanishes as b 2 → −∞ and approaches a finite limit as b 2 → 1.The expression for I 2 simplifies if we set m = 0 (a massless scalar on the brane) and b 2 = 0 (a small boost and / or rotation).Then the sum and integrals can be performed and the behavior for small and large µr can be extracted. 7This leads to 7 Electron self-energy from a bulk graviton The graviton is the most likely candidate for a bulk field.It also provides an interesting contrast to the bulk scalars we have considered so far, since it carries spin and has a non-renormalizeable coupling to the stress tensor on the brane.For these reasons we consider corrections to the electron self-energy induced by a bulk graviton loop.The diagram is shown in Fig. 6.Bulk gravitons in the large extra dimension scenario [10,11,12] have been considered in [13] and we borrow several of their expressions.We expand the 5-D metric about flat space, where k is the 4-D momentum, n is the Kaluza-Klein momentum and we've introduced µ as an infrared regulator.The propagator is written in de Donder gauge, ξ = 1 in the notation of [13].We assume the graviton couples to the 4-D stress tensor on the brane, which leads to the vertex 1 Fermion self energy So far the motion of the brane has only entered in the graviton propagator (69), in a manner exactly analogous to the scalar propagator (29).However the motion of the brane also enters in the effective 4-D coupling.The Newtonian potential on a moving brane was studied by Greene et al. in [9], who found that the relation between the 4-D and 5-D reduced Planck masses becomes 8 For a moving brane r = γR so the reduced 4-D Planck mass is We take this relation to hold in general, i.e. even for a brane that is tilt-like rather than boost-like.
After all these preliminaries we are ready to evaluate the diagram in Fig. 6. 9 The basic diagram is straightforward to write down.
With some Dirac algebra the numerator can be simplified so it is at most linear in Dirac matrices.We Wick rotate, introduce Schwinger parameters, and perform the Gaussian integral over k E .It is convenient to do this in a frame in which the Euclidean vectors have components Then we continue back to Lorentzian signature, restore Lorentz covariance, and expand in powers of p.At zeroth order all terms are either Lorentz invariant or vanish because they are odd under n → −n.At first order in p, after switching from a momentum sum to a winding sum, we find that many terms are either Lorentz invariant or vanish because they are odd under w → −w.Discarding all such terms we are left with a 8 In [9] this was expressed in terms of Newton's constant, G 4 = G 5 /γ2πR with G 4 = 1/8πM Lorentz-violating contribution to the effective Lagrangian, L = ic µν ψγ µ ∂ ν ψ where10 η µν b 2 I gravity (76) We have written (76) as a product of The function I gravity is shown in Fig. 7.It simplifies if we set m = 0 (a massless fermion on the brane) and b 2 = 0 (a small boost and / or rotation).Then the sum and integrals can be performed and the behavior for large and small µ can be extracted.For graviton loops there is no IR divergence, even for a massless fermion on the brane, and we find 8 Scalar self-energy from a bulk graviton Finally we consider corrections to the self-energy of a minimally-coupled scalar field due to a bulk graviton loop.We assume the graviton couples to the 4-D stress tensor on the brane, T µν h µν z=0 (78) The diagram we wish to evaluate is shown in Fig. 8. 11 With the brane-to-brane graviton propagator (69) the basic diagram is straightforward to write down.
Compared to the electron self-energy considered in section 7 the main difference is in the contractions of the stress tensors in the numerator.As is by now familiar we Wick rotate, introduce Schwinger parameters, and perform the Gaussian integral over k E .It is convenient to do this in a frame in which the Euclidean vectors have components Then we continue back to Lorentzian signature, restore Lorentz covariance, switch from a momentum sum to a winding sum, and expand in powers of p.At zeroth order in p the expression is Lorentz invariant.At first order in p the result vanishes because all terms are odd under w → −w.At second order in p many of the terms are Lorentz invariant.Discarding all Lorentz-invariant terms we are left with a Lorentz-violating contribution to the effective Lagrangian, L = 1 2 k µν ∂ µ ϕ∂ ν ϕ where12 η µν b 2 I scalar gravity (82) We have written (82) as a product of • a loop factor 1 16π 2 , • a dimensionless coupling The function I scalar gravity is in Fig. 9.It simplifies if we set m = 0 (a massless scalar on the brane) and b 2 = 0 (a small boost and / or rotation).Then the sum and integrals can be performed and the behavior for large and small µ can be extracted.There is no IR divergence in I scalar gravity , even for a massless scalar on the brane, and we find

Conclusions
In this work we considered a braneworld which is moving or spiraling around a compact extra dimension which we take to be a circle of radius R. The configuration is described by an effective radius r for the compactification and a 4-vector b µ that spontaneously Figure 9: The quantity I scalar gravity appearing in the scalar self-energy due to a bulk graviton loop.The function increases rapidly but has a finite limit as b 2 → 1.
breaks the Lorentz symmetry of the brane worldvolume.
Loops of bulk fields are sensitive to the parameter b µ and can induce Lorentz-violating terms in the 4-D effective action.We explored this, emphasizing the dimension-4 terms which correct the electron self-energy and the electron -photon vertex.
The Standard Model Extension is a general framework for incorporating Lorentz violation and provides many effects to explore.In addition to the QED effects mentioned above, we considered Lorentz-violating corrections to the self-energy of a scalar field, L = 1 2 k µν ∂ µ ϕ∂ ν ϕ with coefficients k µµ given in (66) and (82).Taking the scalar field as a proxy for the Higgs field the experimental bounds on k µν are surprisingly good [6], having reached the level of 10 −12 -10 −20 [15] or 10 −13 -10 −27 [16].
While many similar calculations could be done, there are also theoretical issues worth exploring.In particular it would be interesting to understand soft emission from a moving braneworld.This should be related to the infrared behavior of the diagrams we have considered.For example for µ 2 = 0 the vertex correction (54) has an IR divergence when p 2 1 = p 2 2 = m 2 that should cancel against soft emission in suitable inclusive observables.
Any signal for Lorentz violation in the present epoch would be of the utmost significance.One can also entertain the idea that, although Lorentz-violating effects are extremely small today, they may have been larger in the early universe.Perhaps a braneworld was highly boosted in the early universe and only slowed and stabilized with time.Could the attendant violation of Lorentz symmetry in the early universe leave an observable imprint on cosmology?The bottom part of the envelope is horizontal and intersects the x axis at x = −t (93)

1 Figure 1 :
Figure 1: One-loop electron self-energy arising from a Yukawa coupling to a bulk scalar.

Figure 2 :
Figure 2: The quantity I 1 appearing in the electron self-energy as a function of b 2 .

1 Figure 3 :
Figure 3: One-loop vertex correction due to a bulk scalar.

1 Figure 4 :
Figure 4: One-loop scalar self-energy arising from a cubic coupling to a bulk scalar.

Figure 5 :
Figure 5: The quantity I 2 appearing in the scalar self-energy as a function of b 2 .

1 Figure 6 :
Figure 6: Electron self-energy due to a bulk graviton loop.

) where M 5
is the 5-D reduced Planck mass.The graviton propagator is

9
There is another self-energy diagram at one loop but it does not induce Lorentz violation on the brane.

Figure 7 : 2 Figure 8 :
Figure 7: The quantity I gravity appearing in the electron self-energy due to a bulk graviton loop.The function decreases rapidly but has a finite limit as b 2 → 1.

Figure 10 :Figure 11 :
Figure 10: The case of light-like b µ .The angle between the brane and the X axis is θ = tan −1 λ.The brane moves along the Z axis with velocity λ; the perpendicular component of the velocity is denoted β.