Promises and challenges of high-energy vortex states collisions

In quantum field theory, the wave function of a plane wave one-particle state of a scalar field with three-momentum ${\bf k}$ and energy $E=\sqrt{M^{2}+{\bf k}^{2}}$ can be written as

We assume that the state is normalized by one particle per large but finite volume $V$: $\int_{V}d^{3}r\,j^{0}=\int_{V}d^{3}r\,|\psi_{\rm PW}|^{2}=1$, where $j^{\mu}=i\Psi^{*}\partial_{\mu}\Psi+c.c.$ is the conserved current. This leads to the plane-wave normalization factor $N_{\rm PW}=1/\sqrt{V}$. It should be mentioned that there exist well-known limitations of the one-particle wave function for extremely localized states as well as other related subtleties to the proper normalization of states, [32]. We do not discuss them here, as they will not affect the main physics points we cover in this review, although they may be quantitatively important in processes with non–plane-wave states. Also, the cross sections calculations will not depend on the normalization coefficients if one uses the same normalization for scattering matrix elements, the flux, and the phase space density of the final states.

The simplest type of a vortex state is the so-called Bessel state [33, 34]. This is a monochromatic solution of the wave equation constructed from plane waves with the same longitudinal momentum $k_{z}$, the same modulus of the transverse momentum $|{\bf k}_{\perp}|=\varkappa$ (sometimes called the conical transverse momentum), and, as a result, the same energy $E$. Its wave function in the cylindrical coordinates $(\rho,\varphi_{r},z)$ can be split into the trivial time and $z$-dependent part and a non-trivial transverse wave function:

where $N_{\rm Bes}$ is the new normalization coefficient. Following [33, 34], we write the transverse part of the wave function as

Here $\ell$ is an integer number defining the phase with which every plane wave enters this superposition. Performing the integration, we get

Fig. 1.1 offers a visualization of the Bessel vortex state in coordinate space.

The all-important phase factor $\exp(i\ell\varphi_{r})$ highlights the presence of a phase vortex around the phase singularity line. The “order” of the phase vortex is quantified with the topological charge. To define it, take the phase of the wave function $\arg\psi({\bf r})$, compute its gradient ${\bm{\nabla}}[\arg\psi({\bf r})]$, integrate it along a closed path around the singularity line, and divide by $2\pi$. The result, $\oint d{\bf r}\cdot{\bm{\nabla}}[\arg\psi({\bf r})]/2\pi$, is an integer number called the topological charge. This integral does not depend on the path (provided the path encircles the singularity line exactly once) and is equal to $\ell$ for the wave function in Eq. (2.4). Moreover, even if one “disturbs” the vortex state by adding an smooth wave function without a phase singularity, the topological charge remains the same: it is said that the phase vortex is topologically protected.

Since the vortex state is an eigenstate of the OAM $z$-projection operator $\hat{L}_{z}=-i\hbar\partial/\partial\varphi_{r}$, it carries the OAM $z$-projection $L_{z}=\hbar\ell$. Due to the phase factor $\exp(ik_{z}z+i\ell\varphi_{r})$, the wave fronts are represented by helicoidal surfaces. The streamlines orthogonal to these surfaces show the local current density ${\bf j}({\bf r})$. The concentric rings of intensity, defined by the Bessel function $J_{\ell}(\varkappa\rho)$ and shown in Fig. 1.1, are also a characteristic feature of the vortex state. For $\ell\not=0$, $J_{\ell}(0)=0$, so that the intensity is exactly zero on the axis; near the axis, $\psi_{\varkappa,\ell}\propto(\varkappa\rho)^{\ell}$, and the first maximum of the intensity is located at $\rho\approx|\ell|/\varkappa$.

In momentum space, the Bessel state is represented by an infinitely thin circle with radius $\varkappa$ located in the transverse plane at $\langle{\bf k}\rangle=(0,0,k_{z})$, see Fig. 2.1. The momenta of the plane wave components of the Bessel state cover the surface of a cone with the opening angle $\theta$ defined as $\tan\theta=\varkappa/|k_{z}|$. The conical transverse momentum $\varkappa$ shows the transverse momentum content hidden inside the Bessel vortex state. The paraxial regime, to which we will often refer below, corresponds to $\theta\ll 1$.

Although the height of the intensity rings given by $J^{2}_{\ell}(\varkappa\rho)$ decreases as $\rho$ grows, the exact Bessel state cannot be viewed as localized near the axis. Without a proper radial regularization, the Bessel state is non-normalizable in the transverse plane. Using the same normalization condition, one particle per large cylindrical volume of length $d_{z}$ and radius $R$, we get

Thus, the Bessel state normalization coefficient is $N_{\rm Bes}=\sqrt{\pi/(d_{z}R)}$. For the extreme case of $|\ell|/(\varkappa R)$ close to 1, the normalization condition is modified, see [35].

Just like plane waves, the Bessel states form an orthogonal basis in the transverse plane:

By inserting (2.3) in this condition, one can deduce the regularizing prescription for the square of the radial momentum delta function [33, 34]:

The fact that Bessel states are non-normalizable in the transverse plane may lead to singularities and other artifacts in cross section calculations. A possible way to render the expressions regular is to introduce smearing in $\varkappa$ by the amount $\sigma_{\varkappa}$ [36]. This can be done with the aid of a properly normalized function $f(\varkappa)$ which is peaked at $\varkappa=\varkappa_{0}$ and quickly decreases at $|\varkappa-\varkappa_{0}|>\sigma_{\varkappa}$. One then defines the smeared vortex state

where $k_{z}$ can either be kept fixed or vary together with $\varkappa$ if one insists on a fixed $E=\sqrt{M^{2}+k_{z}^{2}+\varkappa^{2}}$. In this way, one effectively regularizes the vortex state in the transverse plane; the wave function now exhibits only a few (of the order of $\varkappa_{0}/\sigma_{\varkappa}$) intensity rings and then quickly decreases. A drawback of this approach is that results are sensitive to the choice of the function $f(\varkappa)$.

As for the final state particles, one has freedom to choose a basis for their description. The usual choice for outgoing particles is the plane wave basis, which conveniently implements momentum conservation. Whenever one discusses the final state angular distribution, one automatically assumes the plane wave basis. However, one could also describe one or several final states particles in the basis of Bessel states. When doing so, one first must decide with respect to which axis one defines the final particle vortex states. A natural choice is the same axis $z$ as for the initial state, together with the assumption that they reside in the same large normalization volume of the initial Bessel state. Then, one just needs to replace the plane wave number of states with its Bessel state counterpart according to the following rule [34, 15]:

Here, $\Delta\ell^{\prime}$ is a reminder that all final state OAM values must be counted. If one chooses a different axis $z^{\prime}$ with respect to which the final Bessel states are to be defined, the expression for the number of states remains the same but much care must be taken in order to account for two mismatching axes. In this case, as described in [36, 37], the exact Bessel states lead to non-physical artifacts, which can be eliminated once smearing (2.8) is taken into account.

Although the exact Bessel states are convenient for scattering amplitude calculations, they lead to pathologies when integrating the cross section of a two Bessel states collision over the final phase space, see Section 3.3. Regularization of these artifacts with the aid of smearing procedure such as (2.8) introduces certain arbitrariness. A more physically appealing choice is to use other vortex states such as the Laguerre-Gaussian (LG) modes, which contain the all-important $\exp(i\ell\varphi_{r})$ factors and are normalized in the transverse plane. One can also impose the Gaussian profile on the longitudinal distribution, which renders the LG vortex state well localized in three dimensions. Such a state cannot be monochromatic and, therefore, evolves in time, which is also a realistic feature.

A LG wave packet takes the simplest form in the paraxial approximation, when the typical transverse momenta inside the wave packet are much smaller than the average longitudinal momentum and the mass. A description of the LG wave packets in a form convenient for high-energy collisions of massive particles was developed in [32]. Adapting the Lorentz-covariant formalism developed previously for neutrino wave packets [38], Ref. [32] begins with a relativistic Gaussian wave packet in momentum space:

Here, the four-vector $k$ satisfies the on-mass-shell condition $k^{2}=M^{2}$, which implies $E=\sqrt{M^{2}+{\bf k}^{2}}$, $\bar{k}$ is the mean four-momentum, which also satisfies $\bar{k}^{2}=M^{2}$, the parameter $1/\sigma$ shows the spatial extent of the wave packet, and $K_{1}$ is the modified Bessel function. This wave packet satisfied the Klein-Fock-Gordon equation and is normalized with the Lorentz-invariant normalization measure

Its expression in coordinate space and various properties were also presented in [32]. The exact relativistic Gaussian wave packet can be simplified in the paraxial approximation, which is expected to apply to most situations and which, as argued in [32], should be defined as $1/\sigma\gg\lambda_{c}$, the Compton wavelength of the particle with mass $M$. In this approximation, we again choose axis $z$ along the mean three-momentum ${\bf k}$, so that $\bar{\bf k}=(0,0,\bar{k}_{z})$, and represent the momentum space wave function as

which is again normalized according to (2.11). Here, we $1/\sigma_{\perp}$ and $1/\sigma_{z}$ show the transverse and longitudinal spatial extents of the wave function. For the wave packet which is spherically symmetric in its rest frame as implied by (2.10), one finds $\sigma_{z}=\bar{\gamma}\sigma_{\perp}$, where $\bar{\gamma}=\bar{E}/M$, $\bar{E}=\sqrt{\bar{\bf k}^{2}+M^{2}}$. This result agrees with the intuition of the relativistic contraction of longitudinal distances. However one could also imagine the situation where the longitudinal and transverse extents are independent parameters determined by different elements of an experimental setting. In particular, by making $\sigma_{z}$ arbitrarily small, one can obtain a close approximation to the Gaussian beam instead of a wave packet.

Next, Ref. [32] presents the LG wave packets, which were first constructed in the Lorentz-covariant form, then in the paraxial approximation, and finally nonparaxial corrections were investigated. In particular, in the paraxial approximation, the principal mode LG wave packet can be written as

which is also normalized according to (2.11). Notice that one recovers the Gaussian wave packet by setting $\ell=0$.

The coordinate space wave functions for the Gaussian and LG wave packets can also be explicitly found. As expected, they exhibit a characteristic time evolution. At early times the wave packet is broad, but as it moves along axis $z$ with the average velocity $u=\bar{k}_{z}/\bar{E}$ it shrinks and at some instant reaches the minimal dimensions of $1/\sigma_{\perp}$ and $1/\sigma_{z}$. After this momentary focusing, it broaden again. Notice that, when considering collision of two such wave packets, one must take into account not only their shapes but also the time difference $\Delta\tau$ between the instants of their focusing and the distance ${\bf b}$ between the focal points. All these offset parameters can be taken into account with the additional factor $\exp(ib^{\mu}k_{\mu})$ in one of the wave functions, where the four-vector $b^{\mu}=(\Delta\tau,{\bf b})$.

The full set of LG solutions contains the second parameter, $n$, so that the radial wave function exhibits $n+1$ radial intensity maxima. The expression (2.13) represents only the principal LG modes with $n=0$. The paraxial expressions for the general LG wave packets of relativistic massive particles were also obtained in [32], and their properties were studied. Description of the Gaussian and LG vortex wave packets was further developed in [39, 40] within the relativistic Wigner function representation. For a pedagogical discussion of the Wigner functions applied to particle collisions, see [41]. Wigner functions for the Gaussian and LG vortex wave packets were given in the paraxial approximation and then used to derive a universal expression for generic $2\to 2$ cross sections in collisions of two Gaussian wave packets or a LG state with a Gaussian state, which we will review in Section 3.4.

Before proceeding further, it is instructive to discuss qualitative features of vortex states which may at first appear confusing.

A convenient property of a plane wave that it remains a plane wave in any inertial reference frame, albeit with transformed energy and momentum. Bessel vortex states do not follow this pattern. To construct a vortex state, one first needs to fix a reference frame and then to select an axis which will be the phase singularity axis. For the Bessel vortex state, a longitudinal boost will keep it a monochromatic Bessel state with a the same $\varkappa$ and $\ell$ but different $k_{z}$ and $E$. For a Laguerre-Gaussian state, a longitudinal boost makes the vortex beam non-monochromatic, as the focal point will itself be moving; however, for paraxial beams, this effect is minor. Performing a transverse boost changes the space-time evolution significantly even for Bessel vortex states. Still, the momentum space wave function retains the ring structure with a non-trivial phase profile. Such states were dubbed spatiotemporal vortices [42, 43]; they will be briefly discussed in Section 2.6. Thus, when dealing with vortex states and especially with their collision, we unavoidably break the Lorentz covariant description of the initial state.

The dispersion relation $E^{2}=\varkappa^{2}+k_{z}^{2}+M^{2}$ of course holds for every individual plane wave component of a vortex state. However, the relation between the energy and the average momentum $\bar{\bf k}=\langle{\bf k}\rangle=(0,0,k_{z})$ is modified. For the monochromatic Bessel state with energy $E$, we obtain $E^{2}\not=\bar{\bf k}^{2}+M^{2}$. One can discuss this modification in terms of dynamical correction to the mass of the particle [32, 44].

Next, focusing on the orbital angular momentum, one must distinguish the extrinsic and intrinsic OAM. Consider a freely propagating localized wave packet with an average momentum $\bar{\bf k}$. In a chosen coordinate frame, the motion of its centroid is described with $\langle{\bf r}\rangle$. The two vectors can be used to construct the quasiclassical OAM ${\bf L}_{\rm ext}=\langle{\bf r}\rangle\times\bar{\bf k}$. This is the extrinsic OAM, and it can be set to zero by a shift of the origin of the coordinate frame. By construction, ${\bf L}_{\rm ext}\perp\bar{\bf k}$.

The OAM associated with the vortex state is intrinsic. It is generated by the phase vortex and should be defined with respect to the built-in phase singularity axis, which is not related to the coordinate frame choice. For the scalar field, by construction, it is parallel to the average momentum: $\langle{\bf L}\rangle_{\rm int}\,\|\,\bar{\bf k}$, see Fig. 2.1. For an optical field, the full picture is more involved and allows for several forms of the OAM [45].

A parallel shift of the reference axis does not shift the average intrinsic OAM of a vortex state but it broadens its OAM spectrum. Indeed, select, in addition to axis $z$, another axis $z^{\prime}$, which is parallel to $z$ but shifted in the transverse plane by the impact parameter ${\bf a}_{\perp}=(a\cos\varphi_{a},a\sin\varphi_{a})$. Consider a scalar Bessel state with parameters $\varkappa$ and $\ell$ defined with respect to $z^{\prime}$. Then it can be expanded in the basis of Bessel state defined with respect to axis $z$ as (see e.g. Appendix C of [46])

The value of $\varkappa$ is preserved, while the values of $\ell^{\prime}$ differ from $\ell$ by about $\varkappa a$. If the shift parameter $a$ is not too large, this OAM range stays narrow. However, if $z$ and $z^{\prime}$ are separated by a macroscopic distance, this distribution will be extremely broad.

One can also define yet another axis $z^{\prime\prime}$, which is tilted by angle $\alpha$ with respect to the axis $z$, and define a Bessel state around $z^{\prime\prime}$. The expansion of the new Bessel state via the old basis of Bessel state around $z$ will now involve an infinite sum of all $\ell^{\prime}$ [36]. This is hardly surprising; even a plane wave at oblique incidence can be expanded in the Bessel state basis as [47]

This extreme broadening of the OAM spectrum from a single $\ell$ to an infinitely broad $\ell^{\prime}$ range is an artifact of the Bessel states and can be regularized with smeared-Bessel or with LG states [36]. However the important lesson is that if one tries to reveal the intrinsic OAM of a vortex state, one must use the appropriate axis with respect to which to measure it, the phase singularity axis.

This has important consequences for experiment. Suppose we know that a vortex photon is produced but we do not know where its phase singularity axis points to. When a vortex photon hits a traditional pixelized detector, it collapses, in the usual quantum-mechanical sense, to a localized state, which erases the information about its wave function and a possible phase vortex it may have had. No OAM can be detected in this way, even if the photon was in a vortex state. These arguments were recently developed in [48, 49] in a more formal way in terms of the strong postsection protocol which is always implicitly applied when we discuss measurements with traditional local detectors, see more discussion in Section 5.2.2.

One can, however, detect the photon in a way which coherently probes a spatial patch of its wavefront of a certain transverse extent. This can be done, for example, by letting it pass through a customized diffraction grating before hitting the detector. This is how OAM sorters work for optical photons and electrons [1, 11, 12, 50, 8]. However this device will be able to detect the phase vortex of the incident light only if the phase singularity axis directly passes through the device. If it does not, a small local patch of the vortex state wavefront far away from the phase singularity axis will resemble a plane wave with an additional transverse momentum $\ell/a$. Again, no OAM can be detected in this way.

To illustrate the point, suppose a pointlike source emits a spherical wave with $L=1,L_{z}=1$ state with respect to axis $z$. The outgoing wave function $Y_{1}^{1}(\theta,\varphi)\,e^{ikr}/r$ contains the spherical harmonic $Y_{1}^{1}(\theta,\varphi)\propto\sin\theta e^{i\varphi}$, which exhibits a phase vortex with respect to axis $z$. When the outgoing wave reaches a traditional pixelized detector, it leads to the particle detection event at any non-zero polar angle $\theta$. However, the fact that, in this particular event, the particle hits the detector at angle $\theta$, does not mean that the quantization axis changes its direction. It always points along axis $z$, not in the direction of $\theta$. In order to detect the presence of the phase vortex, one would need to focus on the very forward or very backward regions of small $\sin\theta$ and employ an OAM sorter, see, for example, a similar discussion of photons emitted by atoms [51].

In that sense, detecting the OAM state is very different from detecting helicity or the spin state of a particle. When measuring spin, we can use any convenient quantization axis, for example, the direction in which the particle hit a local detector. However we do not have this freedom when measuring the OAM: if a wave, prior to detection, contained a phase vortex, it was defined with respect to the quantization axis defined by the emission process itself. If one still insists on detecting the vortex state and inserts an OAM sorter in a detector of finite size, then a certain spectrum of the OAM states can indeed be observed. But if the phase vortex does not go through the aperture of the detector, the resulting OAM spectrum will not be indicative of the OAM in the incident wave. It is a manifestation of the mere fact that even a plane wave with oblique incidence can be expanded in the basis of Bessel states as in Eq. (2.15).

These remarks are especially relevant for vortex photons which can be emitted in a process that we have no control of. For example, it was theoretically predicted that a Kerr black hole twists the wavefront of a photon emitted nearby [52]. Every vortex photon leaving the Kerr black hole vicinity will posses a phase vortex with respect to the phase singularity axis. However there is no chance of detecting its OAM with telescopes, as it would only be feasible under the unrealistic condition that the phase singularity axis passes straight through the telescope aperture. Equipping a telescope with OAM diagnostic instrumentation such as vortex masks will simply impart an instrument-induced OAM on the wave front, not helping to detect the original OAM of the photon. Multipoint interferometry suggested in [53], too, is of use only in the vicinity of an optical vortex. If the phase singularity axis is light years away, its presence cannot be detected locally.

It must be mentioned that the recent works [54, 55] claim to have detected the OAM structure of the radio emission from the black hole in M87 collected by the Event Horizon Telescope. This report is hard to reconcile with the above discussion.

Another example where similar concerns apply is the recent suggestion of [56] that photons and charged particles produced in non-central heavy ion collisions are in fact twisted by the strong local magnetic fields. They may well be twisted, but only with respect to a specific axis defined by each non-central collision event. The photons and charged particles are emitted at any angle with respect to this axis, and when they hit a local detector, they do not reveal their twisted state. Thus, it remains a major challenge to devise an experiment which could detect the OAM of final particles in such collisions.

Historically, the physics of vortex states began with vortex laser beams in specific LG modes [19]. However, to illustrate the principles of vortex photon construction in potential application to high-energy collisions, we limit the discussion below to the Bessel photon states, which are exact monochromatic solutions of the Maxwell equations applicable within and beyond the paraxial approximation. Bessel photons were introduced in [57, 58]. Although it is possible to quantize the electromagnetic field directly in the space of Bessel photon states [59], this is not needed, as one can define Bessel photons as superposition of plane wave photon states.

The first high-energy process with vortex photons studied theoretically was Compton backscattering in which a vortex optical photon collides with a plane wave ultrarelativistic electron [33, 34]; that work set off exploration of vortex states in particle physics. A review of properties of vortex photons convenient for particle physics calculations can be found, for example, in [8]. The discussion below follows these references.

It is well known that, in quantum field theory, the operator of the total angular momentum (AM) $\hat{{\bf j}}$ commutes with the Hamiltonian, while the spin and OAM operators do not. Therefore, it is possible to construct one-particle states which are eigenstates of $\hat{j}_{z}=\hat{s}_{z}+\hat{L}_{z}$ but not of $\hat{s}_{z}$ and $\hat{L}_{z}$ individually. Put simply, a vortex photon cannot be, strictly speaking, in a state of definite OAM. This feature can be recast in the form of spin-orbital interaction in free propagating light and it was extensively explored by the optics community [60]. Thus, when we say “vortex photon”, we do not mean that it is an eigenstate of $\hat{L}_{z}$. We only mean that the OAM values involved in its construction are nonzero and potentially large. However, as explained already in 1992 [19], within the paraxial approximation when the cone opening angle $\theta_{k}\ll 1$, one can talk about approximately conserved $s_{z}$ and $L_{z}$, in the sense that the photon is mostly in the state of a single $L_{z}$ and that its $L_{z}$ evolution along the $z$ direction can be neglected within typical longitudinal scales of the problem.

To construct the exact Bessel one-photon state, we begin with a monochromatic plane-wave electromagnetic field with helicity $\lambda=\pm 1$, which is described in the Coulomb gauge by

with the familiar plane-wave normalization factor. The polarization vector is orthogonal to the wave vector: ${\bf e}_{{\bf k}\lambda}\cdot{\bf k}=0$. Quantization of this field produces plane wave photons with momentum ${\bf k}$. This state is an eigenstate of the helicity operator $\hat{\Lambda}=\hat{{\bf s}}\cdot{\bf n}_{k}$, where ${\bf n}_{k}={\bf k}/|{\bf k}|$.

Let us now fix a reference frame, select axis $z$, and construct the Bessel photon with helicity $\lambda=\pm 1$ and the total angular momentum $j_{z}=m$ with respect to axis $z$ as a monochromatic superposition of plane waves with fixed longitudinal momentum $k_{z}=|{\bf k}|\cos\theta_{k}$, fixed modulus of the transverse momentum $\varkappa=|{\bf k}_{\perp}|=|{\bf k}|\sin\theta_{k}$, but arriving from different azimuthal angles $\varphi_{k}$. The usual dispersion relation holds for every plane wave component: $k_{z}^{2}+\varkappa^{2}=\omega_{{\bf k}}^{2}$. Using the Coulomb gauge for all plane wave components, we get

Notice that the Fourier amplitude $a_{\varkappa m}({\bf k}_{\perp})$ is the same as in the scalar case (2.3) but it now depends on the conserved total angular momentum $z$-projection $m$, not on the OAM. However, being a scalar quantity, $a_{\varkappa m}({\bf k}_{\perp})$ is an eigenfunction of both $\hat{j}_{z}$ and of $\hat{L}_{z}=-i\partial/\partial\varphi_{k}$ with the same eigenvalue $m$.

The polarization vector ${\bf e}_{{\bf k}\lambda}$ inside the integral (2.17) depends on ${\bf k}$ and cannot be taken out of the integral. It means that the polarization state of a vortex photon must be described, in coordinate space, with a polarization field rather than a fixed polarization vector. Since each plane wave component of the vortex photon comes with its own polarization vector, it is convenient to describe it in a uniform fashion with respect to axis $z$. Namely, define spin $z$-projection operator $\hat{s}_{z}$ and construct its eigenvectors ${\bm{\chi}}_{\sigma}$: $\hat{s}_{z}{\bm{\chi}}_{\sigma}=\sigma{\bm{\chi}}_{\sigma}$, where $\sigma=\pm 1,0$. There explicit form is

Now, the polarization vector of any plane wave component can be expanded in the basis of ${\bm{\chi}}_{\sigma}$ with the aid of Wigner’s $d$-functions [61]:

The first, second, and third rows and columns of this matrix correspond to the indices $+1,\,0,\,-1$. Performing the summation in Eq. (2.19), one gets explicit expressions for the polarization vectors:

Although the vectors ${\bf e}_{{\bf k}\lambda}$ are not eigenstates of $\hat{s}_{z}$ and $\hat{L}_{z}$ separately, they are eigenvectors of $\hat{j}_{z}$ with zero eigenvalue. Thus, when the operator $\hat{j}_{z}$ acts on (2.17), it extracts the value $m$ from $a_{\varkappa m}({\bf k}_{\perp})$ alone. Finally, performing the angular integration in Eq. (2.17), one can compactly write the Bessel photon in cylindrical coordinates:

The description of the Bessel photon simplifies in the paraxial approximation $\theta\ll 1$. In this case, ${\bf e}_{{\bf k}\lambda}\approx e^{-i\lambda\varphi_{k}}\,\bm{\chi}_{\lambda}$, so that the Bessel photon now possesses not only a definite value of $j_{z}=m$ but also (approximately) conserved values of $s_{z}=\lambda=\pm 1$ and $L_{z}=m-\lambda$. Notice however that this expression cannot be extended to the exact forward limit, as the factor $e^{-i\lambda\varphi_{k}}$ becomes undefined.

Anticipating vortex particle collision kinematics, we remark that a counter-propagating vortex photon defined with respect to the same axis $z$, can be described by the above expressions assuming that $k_{z}<0$ and replacing $m\to-m$ in the expression for the Fourier amplitude $a_{\varkappa m}$. The expression for the polarization vector (2.20) stays unchanged, but $\cos\theta<0$. The paraxial limit is now given by $\theta\to\pi$, in which case ${\bf e}_{{\bf k}\lambda}\approx e^{+i\lambda\varphi_{k}}\,\bm{\chi}_{-\lambda}$.

The above formalism can be immediately extended to a massive spin-1 particle of mass $M$ described with the polarization vector $V_{\mu}(\lambda_{V})$. The only modifications are that the dispersion relation changes to $k_{z}^{2}+\varkappa^{2}+M^{2}=E^{2}$, and that the third polarization state with $\lambda_{V}=0$ is now available. The orthogonality condition holds for each plane-wave component with four-momentum $k_{\mu}$ inside the vortex state: $k^{\mu}V_{\mu}(\lambda_{V})=0$. For the transverse polarization states with $\lambda_{V}=\pm 1$, the expression (2.20) applies as it stands. The longitudinal polarization now includes the time-like component and is described by the four-vector $V_{\mu}(\lambda_{V}=0)=\gamma(\beta,{\bf n}_{k})$, where $\gamma$ and $\beta$ are the standard relativistic quantities. One can express the space-like part of this four-vector in the same basis ${\bm{\chi}}_{\sigma}$:

In this way one constructs a Bessel vortex state of a massive vector particle with arbitrary helicity $\lambda_{V}$ and total angular momentum projection $m$.

Although cylindrical solutions of the Dirac equation exhibiting a phase vortex were described as early as 1980 [62, 63], the burst of experimental and theoretical activity on vortex electrons was triggered by [22]. In this paper, vortex electrons were proposed as a novel physical system with rich dynamics, both in free space and in external electromagnetic fields. Several methods to produce vortex electrons were also proposed in [22], and a few years later, they were experimentally demonstrated [23, 24, 25, 64, 65, 66]. During the past decade, the field matured and found various applications, which are covered in reviews [10, 1, 11, 12].

The exact Bessel solutions of the Dirac equation were constructed in [67] and later, within other approaches, in [35, 68, 69, 70, 71, 72]. Our construction of Bessel vortex electrons given below follows [68, 69]; it resembles the construction of Bessel photons and is convenient for computation of vortex electron scattering. It goes without saying that it also applies to other spin-1/2 particles.

We begin with the plane-wave electron with the four-momentum $k^{\mu}=(E,{\bf k})$ and the definite helicity $\lambda=\pm 1/2$:

We define the unit vector ${\bf n}_{k}={\bf k}/|{\bf k}|=(\sin\theta_{k}\cos\varphi_{k},\sin\theta_{k}\sin\varphi_{k},\cos\theta_{k})$, and write the bispinor $u_{k\lambda}$ as in [73]

The bispinors are normalized as $\bar{u}_{k\lambda_{1}}u_{k\lambda_{2}}=2m_{e}\,\delta_{\lambda_{1},\lambda_{2}}$. The spinors $w^{(\lambda)}=w^{(\lambda)}({\bm{n}})$ are eigenvectors of the helicity operator $\Lambda({\bm{n}})=\hat{{\bm{\sigma}}}\cdot{\bm{n}}/2$: $\Lambda({\bm{n}})w^{(\lambda)}({\bm{n}})=\lambda w^{(\lambda)}({\bm{n}})$, with $\hat{\bm{\sigma}}$ being the vector of Pauli matrices. The helicity $\lambda=\pm 1/2$ is the projection of the electron spin onto its propagation direction.

With the choice of spinors $w^{(\lambda)}$ which depend on the azimuthal angle as $\exp(\pm i\varphi_{k}/2)$, the bispinor is defined up to an overall sign and also does not have a well-defined forward limit, see discussion on this issue in [49]. For an alternative, equally possible convention, see [35]. This is a minor inconvenience and it will disappear once we combine these states into the Bessel vortex electron. For now, we only mention that, with this definition, the bispinor $u_{k\lambda}$ becomes also an eigenstate of the total angular momentum operator $\hat{j}_{z}=\hat{s}_{z}+\hat{L}_{z}$ with the zero eigenvalue.

Next, we use this basis of plane wave solutions to construct the Bessel vortex electron state:

This state possesses the definite longitudinal momentum $k_{z}$, the definite conical transverse momentum $|{\bf k}_{\perp}|=\varkappa$, the well-defined helicity $\lambda$ as well as the definite value of the total angular momentum $j_{z}=m$, which must be half-integer. The Fourier amplitude $a_{\varkappa m}({\bf k}_{\perp})$ has exactly the same form as for the scalar case (2.3) but with the half-integer $m$ replacing the OAM value $\ell$. Since $m\pm\lambda$ is an integer number, all components of the bispinor $a_{\varkappa m}({\bf k}_{\perp})\,u_{k\lambda}$ depend on $\varphi_{k}$ as $\exp[i(m\pm\lambda)\varphi_{k}]$ and are well defined.

It must be stressed that, although the Bessel vortex electron (2.25) possesses a well-defined helicity, it cannot be interpreted as a state with a well-defined spin projection on any fixed axis. The helicity operator involves the direction of ${\bf k}$ and it does not correspond to any component of the spin operator.

The OAM and spin are not separately conserved due to the intrinsic spin-orbital interaction within the vortex electron [67]. However, in the paraxial approximation, when $\theta_{k}\ll 1$, this spin-orbital coupling is suppressed, and one deals with two approximately conserved quantum numbers, $s_{z}\approx\lambda$ and $\ell=m-\lambda$.

Just as for scalar fields and photon, one can also construct Laguerre-Gaussian electron states [32]. Properties of these vortex electrons including multipole moments were further investigated in [74, 75].

The exact vortex solutions of the Dirac equation can also be found for an electron moving in the field of an electromagnetic (EM) wave. These solutions can be named Volkov–Bessel states as they extend the well-known Volkov solutions [73, 76] to the Bessel vortex electron. Constructed and investigated first in [35] and later in [77, 78], they offer insights into modifications of the vortex electron properties inside a strong laser field, which is either [35] or linearly [77] polarized. Dynamical properties such the expectation values of the spin and OAM were explored, see also [1]. The numerical analysis of the Bessel electron subject to a strong few-cycle linearly polarized laser pulse reported in [77] demonstrated transverse oscillations of the electron probability density distribution. We also mention in passing that Volkov states of the electron in the field of a vortex electromagnetic wave in plasma were explored in [79].

Vortex solutions of the Dirac equation in Coulomb field represent another class of twisted electron states, which are especially important for computation of twisted electron scattering on heavy atoms [80, 81, 82]. For large nucleus charge $Z$, with the parameter $\alpha_{em}Z\sim 1$, the Born approximation of the scattering amplitude can be unreliable, and one should use the full non-perturbative vortex electron solution valid to all orders of $\alpha_{em}Z$. These states were constructed and discussed in [80, 81]; below we briefly summarize the construction.

In similarity with the free-electron case, one begins with the exact solution of the Dirac equation in the Coulomb potential $\Psi^{(+)}_{km}({\bf r})$, which is composed, asymptotically, by the incident plane wave $\Psi_{k\lambda}({\bf r})$ given by (2.23) and a scattered outgoing wave:

$G^{(+)}_{\lambda}$ is the bispinor scattering amplitude, ${\bf n}_{k}$ and ${\bf n}_{r}$ are the unit vectors in the directions of ${\bf k}$ and ${\bf r}$, respectively. The exact solution with this asymptotic behavior can be constructed as [83, 84]:

Here, $q=(-1)^{\ell+j+1/2}\times(j+1/2)$ is the Dirac quantum number with $j$ and $\ell$ being the total and orbital angular momenta, respectively, $C^{JM}_{j_{1}m_{1}\,j_{2}m_{2}}$ is the Clebsch–Gordan coefficient, $\delta_{q}$ is the phase shift induced by the potential, $D^{J}_{MM^{\prime}}$ is the Wigner rotation matrix [61], and finally $\Psi_{Eqm_{j}}({\bf r})$ is the partial-wave solution of the Dirac equation in the nucleus field which can be found, for example, in [73].

The non-perturbative vortex electron solution in the nucleus Coulomb field is constructed in a similar manner. One looks for the solution which exhibits not the plane wave but the Bessel vortex state asymptotics given by (2.25):

In this expression, a non-zero shift ${\bf b}$ is introduced, which also affects the scattering amplitude. To find the exact solution with this asymptotics, one just uses the same momentum space integral as in (2.25) (with an appripriate ${\bf b}$-shift factor) and applies it to the exact solution (2.27). The resulting exact expressions can be found in [80, 81].

Not only do the vortex photons and fermions carry a non-zero OAM, but they also allow for construction of exotic polarization states, which are impossible for plane wave. A plane wave photon is described by a fixed polarization vector $e^{\mu}$, which is orthogonal to the photon momentum $k^{\mu}$ and is the same everywhere in the transverse plane. Multiplying this polarization vector by a constant phase factor does not lead to any physical consequences. The vortex photon, or in general any non–plane-wave light field, can be decomposed in various plane waves each carrying its own polarization vector $e^{\mu}(k)$ as in (2.17). As a result, the polarization state of the vortex photon is described not with a polarization vector but with a polarization field, in which the Stokes parameters continuously change in space and even exhibit singularity lines. Investigation of the three-dimensional structure of the optical polarization field is the subject of singular optics [85, 86, 50]. Beyond optical range, a mechanism of generation of vortex X rays with exotic polarization states, including the so-called “star” Poincaré beams, was recently proposed for the free-electron laser facilities in [87].

A similar freedom exists for vortex fermions. When constructing the vortex electron, one accompanies each plane wave with the corresponding bispinor $u_{s}(k)$ as in Eq. (2.25). Again, the resulting fermion field is described with a polarization field, which can be defined as the local spin vector smoothly depending on coordinates. Fig. 2.2 shows a few examples of spin-orbit coupled states of a vortex fermion constructed as superpositions of $s_{z}=+1/2$, $\ell=0$ and $s_{z}=-1/2$, $\ell=+1$ states with various relative coefficients. Since the weights of the $s_{z}=+1/2$ and $s_{z}=-1/2$ states are equal in this example, the state is polarized in the transverse plane, with the spin orientation shown by the arrows depending on the azimuthal angle $\phi$. Other polarization states of vortex fermions can also be constructed.

This freedom in preparing customized polarization states of vortex photons and fermions is unthinkable for plane wave states. It represents a new degree of freedom, in addition to the OAM, which can be imposed on the initial particles in vortex state collisions. However, not much has been explored so far, at least with possible particle physics applications in mind. In [28, 29], experimental progress on generation of slow vortex neutrons was reported based on a scheme which relies on various spin-orbit coupled states of vortex neutrons. In theoretical analyses of vortex scattering processes, this opportunity is sometimes mentioned, see e.g. [15], but no systematic analysis exists of the effects to be expected. In fact, in most cases, the vortex photons or electrons are considered as unpolarized. A rare exception is the vortex muon decay calculations [88, 89] where the electron spectrum and azimuthal distribution show significant dependence on the polarization state of the muon. The full potential of this degree of freedom still awaits exploration.

Even defining the unpolarized vortex photon or electron, or the unpolarized cross section with vortex particles, is not a trivial issue. When dealing with an unpolarized plane wave cross section, we just sum over final helicities and average over the initial ones. Since the spin and coordinate dependences are factorized for plane waves, no ambiguity arises. In contrast, the exact expression for the vortex photon or fermion contains two relevant quantum numbers: the helicity $\lambda$ and the total AM $m$. When averaging over $\lambda=\pm 1$, what should we keep fixed: the total angular momentum $m$ or the combination $m-\lambda$, which would correspond in the paraxial limit to the OAM? There is no unique answer to this question. It will eventually depend on the vortex photon or fermion preparation scheme and, in particular, how well the paraxial approximation holds. If one creates twisted photons or electrons by letting an unpolarized plane wave pass through a fork diffraction grating which imposes a given OAM $\ell$ immediately behind the aperture, the intrinsic spin-orbital interaction will lead to $\ell$ evolution downstream the beam, and the polarization state becomes non-trivial at the focal spot. This spin-to-orbital conversion was already experimentally verified in optical vortex beams [90]. Also, an electron microscope can act as an efficient spin polarizer if one prepared a Bessel vortex electron and postselects it near the axis [91]. In any case, as was recently found in [92, 93], different definitions of unpolarized vortex electron and photon cross sections will lead to significantly different predictions for the polarization state of a produced resonance, once high-energy vortex $e^{+}e^{-}$ or $\gamma\gamma$ collisions become possible.

We close this section with the remark that vortex states with a definite OAM or total AM value represent just one example of a broader class of beams or wave packets with custom-tailored wave functions. Let us briefly mention other options considered in literature and demonstrated experimentally.

First, one can take a superposition of vortex states with the same intensity distribution but with the opposite OAM $z$-projection values. The coordinate wave function $|+\ell\rangle+|-\ell\rangle\propto\cos(\ell\varphi_{r})$ leads to the intensity distribution with multi-petal azimuthal angular dependence. When such states scatter head-on, they lead to an azimuthally dependent signal which was absent in the single-OAM case [15, 94]. Behavior of such states of electrons in external magnetic fields was explored theoretically in [95, 96] and verified in experiment in [97]. The spin-orbit coupled states of vortex photon or fermion with an azimuthally-dependent polarization field, which were discussed above and shown in Fig. 2.2, represent yet another example of superposition of different OAM.

In the simplest case, the non-trivial intensity distribution of an OAM superposition state does not evolve downstream the beam. However it is also possible to prepare fine-tuned OAM superpositions which exhibit such evolution in free space, without any external fields. Very recently, the so-called coiling free electron waves were theoretically explored and experimentally produced with the aid of nanofabricated holographic masks in [98]. In these states, the electron density in the transverse plane contains two compact lobes which rotate around each other as the beam propagates, and their angular velocity can be either constant or variable.

One can also consider an electron state with a transverse wave function in the form of a superposition of two compact Gaussian states separated by a controlled distance, which may be larger than the size of the individual Gaussian envelop. Such states were called in [99] Schrödinger’s cat states following the quantum optics tradition. With suitable parameters, scattering of cat states on atomic targets proceeds in a regime which significantly differs from approximate plane wave scattering. In particular, as shown in [99], this process allows one to probe negative values of the Wigner function of the incident wave packet.

Other examples of states with highly non-trivial phase front include Airy beams [100, 101], which have already been demonstrated experimentally for electrons [102], and self-accelerated Dirac particles which are predicted to exhibit remarkable behavior [103]. One can also construct a more exotic form of vortex states called spatiotemporal vortices [42, 43]. In momentum space, these states can also be approximated by a thin ring as for the Bessel state shown in Fig. 2.1. However, now this ring is not orthogonal to $\langle{\bf p}\rangle$ and can even lie in a plane that includes axis $z$. Such states are unavoidably non-monochromatic, thus they exhibit space and time evolution. Experimentally observed for photons in [104, 105], these spatiotemporal vortices are now a vibrant topic on their own.

In broader terms, the concept of wave function engineering for electrons and ions is now being proposed as a novel technique which can expand horizons of traditional microscopy [18, 106]. These states may eventually find applications in particle physics. For example, freely propagating states demonstrating self-accelerating behavior are predicted to exhibit slowing down of an unstable particle decay [103]. Airy beams and other states with non-trivial phase structure may represent an additional probe of the overall phase of the scattering amplitude [107, 108, 41]. Scattering of precisely shaped electron wave packets offers an unprecedented control over the rate, the spatial and the spectral distributions of bremsstrahlung and, potentially, other QED processes [109]. Discussion of these states goes beyond the scope of the present review, but we strongly suggest that particle physics community keeps track of this topic and draws inspiration from the results.

Several schemes for vortex states scattering are possible [1]. They depend on whether the vortex states appear in the initial or final state and on how many particles involved are described as vortex states. Each setting brings its own opportunities and challenges, which we now briefly describe.

Vortex states can appear in the initial or in the final state. From the experimental perspective, these two options are fundamentally different. Experiments with initial state particles in vortex states require new state preparation instrumentation, but the outcome of the collision can analyzed with traditional detectors. However if one aims at producing at least one final particle in a vortex state, one must introduce new instrumentation capable of detecting that a final particle indeed possesses a phase vortex and of measuring its vortex state parameters. This is a major additional experimental challenge.

Indeed, an initial vortex state can be represented as a well-collimated, often paraxial wave packet or a beam. Its vortex state parameters can be analyzed with diffraction gratings or other OAM sorters of small transverse size. In contrast, an outgoing wave function emerging from a typical scattering process has a wide polar angle distribution. If one wishes to detect this outgoing state as a vortex state defined with respect to the collision axis, one would need to coherently detect it in the entire $2\pi$ range of azimuthal angles $\varphi$. At present, no such detectors exist, at least for high-energy particles. As we discussed in Section 2.2, a standard detector instrumentation will be of little use: as long as one detects an outgoing particle in a certain direction, one loses information about its $\varphi$-dependent phase [48, 49]. These challenge may be less severe for processes featuring very narrow angular distributions of one of the final particles, such as Compton backscattering from ultrarelativistic electrons [33, 34].

Next, scattering of a vortex state can proceed in a fixed target or in a collider-like regime. In fixed target scattering, a heavy scattering center can absorb any momentum transfer. Since the vortex initial state is a coherent superposition of plane waves, its scattering amplitude to any final plane wave state can be written is a superposition of individual plane wave scattering amplitudes corresponding to different momentum transfers. This leads to modifications of the differential cross section and the polarization properties of final photons or electrons with respect to the plane wave case. For a pedagogical exposition of wave packet scattering on atoms, see, for example, [110].

These modifications depend not only on the vortex state parameters but also on the location of the scattering center ${\bf b}=({\bf b}_{\perp},b_{z})$ with respect to the vortex axis. For a single scattering center, such as a single atom or an ion in an trap which absorbs a twisted photon, the ${\bf b}$-dependent modifications can be very profound. However, if one considers scattering of vortex states on a macroscopic target with uniformly distributed scattering centers, averaging of the scattering cross section over a range of ${\bf b}$ washes out the azimuthal angle features and renders the results insensitive to the OAM value $\ell$ [111, 68]. However, structures in the polar angle distributions are preserved and can be used to investigate vortex state scattering on atoms. Uniform averaging does not apply to crystals and, in particular, to chiral crystals; as a result, elastic scattering of vortex electron beams can be used to investigate chirality in crystalline materials [112]. Finally, a mesoscopic target of a spatial extent comparable with the waist of the focused vortex beam interpolates between the two cases and allows one to observe residual dependences which were pronounced for the single scattering center but disappeared for a macroscopic target.

In collider-type scattering, where the two initial particles are described as freely propagating wave packets or monochromatic beams, energy and momentum conservation imposes additional constraints on interference of individual plane wave components from the initial state scattering to a given final state. In single-vortex scattering (or single-Bessel scattering if Bessel states are used), a vortex state collides with a counterpropagating plane wave. If one studies the angular distribution of the final particles, the interference disappears, see details below. Decay of a vortex state particle into plane wave final states also follows this scheme.

In contrast, double-vortex scattering, in which two counterpropagating vortex states collide, will exhibit a remarkable interference pattern between different pairs of initial plane wave components scattering into the same final state. Thus, double-vortex scattering gives access to new observables, which cannot be studied with the single-vortex setting. Below we will consider in detail the main kinematic features in these two scattering regimes.