The neutrino’s elusive helicity reversal

The 1957 discovery of parity violation in the weak interactions ¹ swiftly brought attention to two concepts that have become ubiquitous in particle physics: Helicity is the projection of a particle’s spin along its direction of motion, and chirality is the projection of a four-component Dirac spinor onto the sum (positive chirality) or difference (negative chirality) of what in Paul Dirac’s original notation are the “large” and “small” components of the wavefunction for spin-1⁄2 particles (see box 1). Projection onto eigenstates of chirality is central to the standard model of the electroweak interactions. That’s because most weak interactions involve only the negative-chirality components of the interacting leptons and quarks.

For the spin-1⁄2 neutrino, by convention, the helicity and chirality eigenvalues are both normalized to ±1. If the neutrino mass actually were zero, as was presumed before flavor-mixing experiments revealed otherwise in recent decades, helicity and chirality would be equal for neutrinos and opposite for antineutrinos. A massless neutrino would be fully polarized; its state could never be a superposition of two different helicities. A 1958 nuclear-decay experiment carried out at Brookhaven National Laboratory by one of us (Maurice) and coworkers gave a direct indication that the helicity of the neutrino is negative. ² (See the photo on page 42.) That result provided the basis for the standard model’s “left-handed” chiral coupling. (A negative-helicity particle rotates like a left-handed screw, and the coinage “chiral” comes from kheir, the Greek word for “hand.”)

If a massless neutrino starts out with negative helicity, it will exhibit that same helicity for all observers, regardless of their motion. For massless neutrinos, the usual four-component Dirac spinor can be replaced by a two-component Weyl spinor, which would be a purely left-handed eigenstate of both chirality and helicity. And it would be covariant under Lorentz transformation.

But they’re not massless

The 1998 discovery that there must be nonzero neutrino masses ³ has since been extended, with the conclusion that the mass eigenvalues for the three identified types of neutrino must all be different. So at least two of them must be nonzero. For nonzero mass, one can at least imagine going to the neutrino’s rest frame. There its velocity is zero, and hence its helicity is undefined. But by running down a negative-helicity massive neutrino from directly behind and catching up with it, one would see it converted to positive helicity. Its spin direction would not have changed, but its velocity would appear to change sign when the pursuer accelerates to overtake it. Of course, given the very small masses of neutrinos—at least a million times smaller than the electron’s—and their typical MeV or GeV energies in the laboratory, running fast enough is out of the question for any macroscopic pursuer.

What about sending a microscopic particle after the neutrino? That is presumably what various commentators—Martinus Veltman, for example ⁴ —had in mind when they suggested that if the neutrino mass is nonzero, helicity reversal is possible. We discuss three approaches to detecting such helicity reversal. The first method, direct pursuit, is not feasible. But it’s intellectually the most interesting, because helicity reversal of the pursued particle turns out to be protected from observation by a formidable, multilayered shield.

The second approach follows from applying the Dirac equation for a massive neutrino. From that exercise, one concludes that a tiny amplitude for reversed helicity is produced by the standard chiral coupling. Therefore one could, in principle, detect a reversed-helicity component of the neutrino’s wavefunction, present from the moment of its creation. But because the amplitude is so small, that also turns out to be impracticable.

The third approach is to search for neutrinoless double beta decay. That much-sought-after nuclear decay mode requires not only that the neutrino have nonzero mass, but also that the neutrino be its own antiparticle—a so-called Majorana particle (see box 2). The plausibility of this last approach is attested to by the serious experimental undertakings currently under way and planned for that search (see PHYSICS TODAY, January 2010, page 20 ).

Catching up with a neutrino

If we assume that the neutrino we’re trying to catch up with has a rather low energy E, say 100 keV, and its mass m is the biggest that one might infer from neutrino-oscillation experiments, about 10⁻² eV, we get a Lorentz factor γ = E/m = 1/√1 − β ² ≈ 10⁷, or a velocity v = βc ≈ c(1 − 10⁻¹⁴). To catch up with that neutrino, a pursuing particle would require an even larger γ. For an electron, the lightest particle that might be accelerated, that would mean an energy of about 10 TeV, well beyond the capabilities of any electron accelerator currently proposed. A circular accelerator couldn’t do it, because synchrotron radiation losses would make the acceleration prohibitively inefficient. Linear electron accelerators are not subject to that reservation, but linac designs have yet to approach such high energies.

There’s a further difficulty. If a pursuit particle at any speed less than c were moving at a slight angle, θ ≥ 10⁻⁷ radians, to the neutrino velocity v, the component of its velocity along v would be less than the neutrino’s, making pursuit futile. Thus the phase space for successful pursuit is inordinately small, and hence so is the probability of catching up with the neutrino.

That difficulty gets even worse when you take account of the uncertainty principle, which says that the product of the uncertainties in transverse momentum and transverse displacement of either the neutrino or the pursuit particle is at least ℏ/2. One might hope to escape this problem by considering the case of a neutrino traveling at a substantial angle to the velocity of a slower pursuer aimed and timed to intercept it. The pursuer, after all, has a shorter path; it moves along one side of a right triangle while the neutrino takes the hypotenuse.

But now, yet another issue emerges. Suppose the neutrino has a significant transverse momentum (substantially greater than its rest mass) with respect to the pursuit direction. Then, upon catching up, the pursuer sees the neutrino still moving sideways with speed close to c. Therefore the neutrino will still appear to have negative helicity. The probability that the pursuer will measure positive helicity is of order 1/γ, where γ is now (and henceforth) the neutrino’s Lorentz factor as seen by the pursuer.

Rotation and precession

The fact that a neutrino’s helicity is so unlikely to change when it’s run down at an angle depends on a remarkable property of special relativity. As the pursuit particle is accelerated from rest, it sees the neutrino’s velocity vector not only shrinking but also rotating, until eventually it’s orthogonal to the pursuer’s direction. Meanwhile, the acceleration of the pursuit particle also produces an apparent rotation of the neutrino’s spin direction. The differential rotation of the spin direction is dθ′ = β ² dθ/(1 + 1/γ), where dθ is the differential rotation of the neutrino’s velocity direction at each instant. As the pursuer accelerates to catch up, γ decreases, but even its final value is large. So the neutrino’s spin still is almost exactly aligned with the momentum; the helicity is virtually unchanged.

Readers familiar with the Thomas spin precession ⁵ for a particle accelerating with respect to a stationary observer might be skeptical about the argument above for two reasons: First, there is a well-known factor 1⁄2 β ² in the relation between the rotation rates of the spin direction and the velocity direction for a non-relativistic accelerated particle such as an electron in orbit around a proton. The factor 1⁄2 suggests that even for v close to c, the rotation of the spin would be only half as fast as the rotation of the velocity direction. Second, the Thomas spin precession has the opposite sense to the velocity rotation, which might seem to make helicity particularly fragile.

Let’s address those concerns in turn. It’s easy to show that as the speed of the accelerated particle approaches c, the magnitude of the spin-rotation rate approaches that of the velocity, because the relativistic form for the 2 that divides β ² above is actually 1 + 1/γ.

The resolution to the second objection is found by considering carefully what happens to the apparent direction of the neutrino spin as the pursuer accelerates. That’s quite different from Thomas precession in an atomic orbit, where one asks how the spin of a particle in its own rest frame precesses as the particle is centripetally accelerated with respect to an observer. Instead, if we treat the pursuit particle as the observer, we are accelerating that observer while keeping the neutrino’s velocity fixed. Looking at the sequence of Lorentz transformations that imply rotations of the neutrino’s frame, one finds that they are, in fact, the inverses of the set of transformations used to derive the Thomas precession. Thus they closely track the rotation of the neutrino’s momentum direction.

A simpler reason for believing the nearly perfect helicity invariance of a massive neutrino pursued at an angle is that otherwise the limiting case of exact helicity invariance for the massless neutrino would be discontinuous. Thus, if the neutrino’s mass is as small as is generally assumed, only essentially straight pursuit from behind could work, and its phase space for success is impossibly small.

If the helicity of a neutrino were reversed, then for a Dirac neutrino—as distinguished from a Majorana neutrino—the “wrong-helicity” neutrino’s interactions with matter would be negligible. So to see the helicity change, one would have to create a beam of neutrinos with a small wrong-helicity component and observe a tiny quenching of what is, even without that component, a notoriously weak interaction with matter.

But if the neutrino is of the Majorana type, with neutrino and “antineutrino” bundled together in a single, self-conjugate spinor, then the reversed-helicity particle would interact just as strongly, but differently, with matter. Then, if only one could produce the reversed helicity, there would be a way to detect it.

We have, however, just seen that producing the reversed helicity by pursuit is impractical. In addition to all the issues already mentioned, there’s another very practical problem. The reaction cross section for the weak interactions falls with decreasing energy in the center-of-mass reference frame. Thus, even if a pursuit electron were fast enough to catch the neutrino from straight behind, it would still have a rather small velocity with respect to the neutrino. Therefore their interaction cross section would be very small, making detection of effects associated with helicity reversal exceedingly difficult.

It’s worth repeating that in the discussions above, we assumed no force on the neutrino during its motion. We were discussing helicity invariance with respect to the pursuit particle as observer, not helicity conservation in the presence of external forces. A neutrino subjected to a force, or even a torque, could easily violate helicity conservation. For example, if a neutrino carrying a magnetic moment moved through a magnetic field, helicity conservation would be violated. If CP—the eigenvalue for the combined operation of charge conjugation and spatial inversion—is conserved, no mass eigenstate of a Majorana neutrino could have a magnetic moment.6 But magnetic-moment mixing of Majorana mass eigenstates could produce helicity violation.7

Helicity reversal in production

The charged-current weak interactions are well described as having only left-handed chiral couplings. For a massless particle, the projection onto a definite chirality state is the same as the projection onto a definite helicity state. For nonzero mass, however, there’s a difference between the two. An easy way to see that is to consider a process in which an electron is at rest—for example a nuclear transition in which the nucleus emits a neutrino after capturing an inner-shell electron. Evidently the helicity of the electron is undefined, even though the chiral coupling is purely left-handed. By the same token, the massive neutrino created in the transition must have a small component of right-handed helicity. If that neutrino were then reabsorbed in a second reaction in another nucleus, that right-handed component, corresponding to dominantly right-handed chirality, would not interact in the second nucleus—if it’s a Dirac neutrino.

But if the neutrino is a Majorana particle, the right-handed component could conceivably produce interactions characteristic of incident antineutrinos and thus give an apparent neutrino–antineutrino mixing, from which the existence of the reversed-helicity component could be inferred. If one were looking for interactions of the tiny right-handed helicity component in a beam of neutrinos, the beam would have to be ultrapure, with an antineutrino-impurity fraction of no more than 1/γ ². A tall order! The best known experiment of that sort, actually looking for a neutrino impurity in a reactor-produced antineutrino beam, was carried out by Raymond Davis a year before either neutrino variety was actually discovered. ⁸ It had nowhere near the requisite sensitivity to detect such neutrino–antineutrino mixing.

Neutrinoless double beta decay

To infer helicity reversal from a phenomenon other than the nonzero neutrino mass, we are left only with the process of neutrinoless double beta decay, which has yet to be seen—if it occurs at all. Recognized conditions for its occurance are that neutrinos are Majorana particles and that the neutrino mass is nonzero. Conservation of angular momentum in the production or annihilation of a virtual neutrino pair guarantees that the two neutrinos have the same helicity, instead of the opposite helicities one expects for neutrino–antineutrino pairs. Therefore, the observation of neutrinoless double beta decay would constitute evidence for neutrino helicity reversal.

Even that comes with a caveat: There could be lepton-number nonconserving phenomena beyond those associated with massive Majorana neutrinos, meaning that even for Dirac neutrinos, neutrinoless double beta decay might occur through a virtual process involving some new particle of very high mass. ⁹

In summary, then, observable neutrinos always have very high velocity with respect to all components of conceivable detector systems. Therefore, anomalous helicity is very hard to confirm directly, even though the standard model’s chiral coupling implies that there must be a small probability amplitude that a neutrino produced in a weak interaction will have reversed helicity. But helicity reversal might be established indirectly by the observation of the extraordinarily weak, and therefore very sensitive, process of neutrinoless double beta decay—provided that no new interactions outside the standard model’s purview violate lepton-number conservation.