How can helicity be conserved but chirality not?

Question

I read in a book that for $\beta$-decay the electrons have always been found to have an expectation value for their helicity of $h=-v/c$.

Then ist is said in the book, that it follows from this fact that such electrons are in a left-handed chiral state which is characteristic for the weak interaction.

In another article I read that the chiral state of an electron is not conserved in time. The electron will soon evolve a component with a right-handed chiral state and it will be a mixture of right- and lef-handed chiral states.

Suppose after the decay one electron moves like a free particle.
When it evolves a right-handed chiral component in addition to the left-handed component it starts off with, how can its helicity be conserved?

Answer 1

General explanation

Chirality and helicity are in general different quantities. Chirality is connected with the representation of the Lorentz group (left or right) while helicity is connected with projection of spin on momentum direction and becomes to characterize representation only in massless case.

I have meaned following. Particularly (!), representation with spin $s$ of the Lorentz group can be given (in terms of the irreducible representations of the lorentz group) as $\left( s, 0\right) $ (left chiral) or as $\left(0 , s \right)$ (right chiral). These representations are equal only for massive case (there exist an operator which connects them), and the second one is transformed as complex conjugated first one. You may get the second one by acting of spatial inversion or time inversion operators on the first one, and vice versa.

Helicity is the spin projection on the momentum direction. In general, it means that there are $2s + 1$ values of helicity for massive case. But massless case is radically different from massive one. Massless representations are characterized by helicity; There is only one lorentz-invariant (under continuous transformations) helicity value for massless particle (if theory isn't invariant under spatial inversion), and helicity $\lambda $ representation in particular case may be given as $\left( \lambda , 0\right)$, while helicity $-\lambda$ representation may be given as $\left( 0, \lambda \right)$. So only in massless case we may "equalize" helicity and chirality. You may understand this as one of the demonstration of relativistic aberration effect: when the speed of inertial frame is near $c$ the spin vertor "lays down" on the momentum direction independently of it's projections distribution at rest frame. So there is only one value of projection. If you inverse spatial coordinates, you will get the value with minus sign.

So this is a kind of accident that for massless case helicity "coincides" with chirality. Formally they are totally different quantities. For example, gravitons have only two helicities, $2$ (right graviton) and $-2$ (left graviton), similarly to photons, $1$ (right), $-1$ (left). But theory of left photon doesn't coincide with left graviton theory, because they have different helicities.

Let's talk about Dirac spinor case.

Particular case

Let's discuss particular case - Dirac spinor.

Helicity operator is $\hat {h} = \frac{(\hat {s} \cdot \hat{\mathbf p})}{|\mathbf p|}$, while chirality operator is $\hat {c} = \gamma_{5}$. You can show that for massless case $$ \hat{c}u(p) = \hat{h}u(p), \quad p_{0} > 0, $$ and $$ \hat{c}u(p) = -\hat{h}u(p), \quad p_{0} < 0. $$ Here $u(p)$ is the spinor wave: $$ \hat{\Psi} = \sum_{\sigma}\int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi )^{3}2E_{\mathbf p}}}\left( u_{\sigma}(\mathbf p)\hat{a}_{\sigma}(\mathbf p) e^{-ipx} + v_{\sigma}(\mathbf p)\hat{b}_{\sigma}^{\dagger}(\mathbf p)e^{ipx}\right). $$

But in massive case these operators are different. This statement is evident from the fact that helicity is conserved in time but it isn't Lorentz invariant $$ \tag 1 \dot{\hat{h}} = [\hat{h}, \hat{H}] = 0, \quad [e^{\sigma^{\mu \nu}\omega_{\mu \nu}}, \hat{h}] \neq 0, $$ while chirality isn't conserved in time but it is lorentz-invariant: $$ \tag 2 \dot{\hat{c}} = [\hat{c}, \hat{H}]= -2m\gamma_{0}\hat{c}, \quad [e^{\sigma^{\mu \nu}\omega_{\mu \nu}}, \hat{c}] = 0, $$ (so an axial current in electroweak lagrangian isn't conserved) and only in massless case they have identical behaviors under time evolution and Lorentz group transformations.

In terms of previous "paragraph" ("General explanation") these all can be summarized into following statements.

Dirac representation is $\left( \frac{1}{2}, 0\right) \oplus \left(0 , \frac{1}{2} \right)$ (so it is sum of left and right chiral states, they are mixed by mass term). Dirac particle has two possible values of helicity. In massless case we may separate this representation to $\left( \frac{1}{2}, 0\right)$ (left chirality) and $\left( 0 , \frac{1}{2} \right)$ (right chirality) with constant and lorentz-invariant helicities. So we may "equalize" left chirality and helicity $-1$ and right chirality and helicity $+1$.

Axial currents in Standard model

In Standard model's electroweak interactions part you have the expression for charged current in electroweak theory: $$ L_{Ch} = \bar{e}\gamma^{\mu}(A + B\gamma_{5})\nu_{e} W_{\mu} + h.c., $$ or $$ L_{Ch} = J^{\mu}W_{\mu} + h.c., \quad J^{\mu} = \bar{e}\gamma^{\mu}(1 + \gamma_{5})\nu_{e}. $$ Let's get the 4-derivative of axial part $J_{Ax}^{\mu}$ of this current, i.e., part proportional to $\gamma_{5}$: $$ \partial_{\mu}J_{Ax}^{\mu} = (\partial_{\mu}\bar{e}\gamma^{\mu})\gamma_{5}\nu_{e} + \bar{e}\partial_{\mu}\gamma^{\mu}\gamma_{5}\nu_{e} \approx $$ $$ = \left|(\partial_{\mu}\bar{e}\gamma^{\mu}) \approx -m_{e}\bar{e}, \quad [\gamma_{\mu}, \gamma_{5}]_{+} = 0, \quad \gamma^{\mu}\partial_{\mu}\nu_{e} \approx 0\right| = $$ $$ \approx -m_{e}\bar{e}\gamma_{5}\nu_{e} $$ (here $\approx$ means that I have neglected hermitean conhugated summand in $J_{Ax}$ which contracts nonmass terms and doubles mass term).

So, as you can see, an axial component of charged current isn't conserved in time in electroweak processes because electron has mass (or, equivanently, it can "move" from one chiral state to another, as the previous part of answer claims). This conclusion is equal to $(2)$ (that's why I have written it).

But helicity's current is always conserved in time (this fact doesn't depend on mass; this is expressed in $(1)$). You also may build the expression for helicity's weak current and see that it is different from axial current. It can be easily understood because (see first two parts of the answer) in massive case helicity isn't equal to chirality.

So there is nothing strange in the fact that helicity is conserved while chirality doesn't.

Answer 2

Helicity (the correlation of spin and momentum) and chirality (whether a particle couples to the left-handed or right-handed part of the weak interaction) are strongly correlated in the relativistic limit $v\to c$, but not strongly coupled at low speeds.

The electron, following the decay, is happy to believe that it's in its own rest frame and to evolve from a purely left-handed state into a mixture of left- and right-handed components. However, the evolution of its helicity is governed by conservation of angular momentum.

This is, for instance, why $\pi^+\to\mu^+\nu_\mu$ is strongly favored over $\pi^+\to e^+\nu_e$ in charged pion decays, even though the latter would release much more energy. The pion has spin zero, and in the rest frame the decay leptons must have opposite momentum. The weak interaction wants to make a (chirally) left-handed lepton and right-handed antilepton. But to conserve angular momentum you must have (in helicity) to right-handed or two left-handed decay products. The $\mu^+$ is born going lots slower than the $e^+$, so the correlation between spin and helicity is weaker and the muon can be polarized the "wrong way."

Answer 3

I'll clear the confusion up.

First, notwithstanding what's been said to the contrary, chirality is helicity - but only for light speed particles. There is no concept of chirality for sub-light particles. There, it is just helicity. In all cases, it refers to the component of the angular momentum along an axis parallel to the linear momentum. You can see this most clearly in the symplectic version of the Wigner classification (i.e. by looking at the co-adjoint orbits of the Poincaré group, if you want to get technical about it). I linked to the mathematical details of both this and what's to follow later, below, so as not to clutter up the discussion.

The symplectic version of the "Wigner" classification can be applied uniformly both in classical and quantum theory and provides a picture that is more intuitively grounded, while matching what is more commonly called the Wigner classification, in the quantum case, but minus the obfuscation that goes with treatment presented in that case. The picture provided is this:

(1) For sub-light particles (the Wigner class "tardions" or "bradyons"), there are 3 or 4 conjugate pairs of coordinates. Of them, 3 are given, as usual, by $(x,y,z)$ and $(p_x, p_y, p_z)$ for coordinates and momentum.

(1a) For "spin 0", 3 is all there is. The spin 0 modes have spherical symmetry. Correspondingly, this is the sub-family that Newton would have been happy to refer to as "corpuscles", as he had no concept of internal angular momentum.

(1b) Otherwise, the modes only have axial symmetry and there is a 4th conjugate pair that essentially goes with the azimuthal angle $φ$ and the corresponding component of angular momentum $p_φ$. In the quantum setting, the latter corresponds to the quantum number usually written as $m$. Thus, for example, for spin 1, $m ∈ \{-1, 0, +1\}$.

(2) For light-speed particles (the Wigner class "luxons"), there may also be 3 or 4 conjugate pairs of coordinates.

(2a) Again, for "spin 0", there are only 3 and they are as given before.

(2b) The case where it is 4 is less well-studied and has never really received any consistent interpretation in the literature. There are no known elementary particles that fall in that subclass.

(2c) Finally, there is another sub-family, where it is also 3, which I've termed the "helical luxons", or "helion" for short, since they've never actually been given any name that I'm aware of in the literature. For this class - and only this class - there is a concept of a fixed (non-zero) helicity. The component of the internal angular momentum along an axis parallel to the linear momentum is an invariant. It does not change. That invariant corresponds to chirality.

Inclusion in the (2c) class is a pre-condition for anything that interacts with the weak nuclear force, since the weak nuclear charge, itself, is proportonal to left-chirality for matter and right-chirality for anti-matter; and that, in turn, requires that there actually be such an invariant as chirality.

Consequently, only light speed particles can interact with the weak nuclear force. Even the ones that "go at sub-light" are in reality light-speed. More on that below and spoiler alert: Higgs.

Intuitively speaking, the reason the internal angular momentum along an axis parallel to linear momentum is not invariant for sub-light particles is simply because motion is relative. More to the point: it's relative in the sense that you can overtake a sub-light particle. So, while it may be traveling away from you, in one frame of reference; in another frame of reference that's moving fast enough to overtake it, it's going in the opposite direction.

Consequently, what appears in your frame of reference as rotation counter-clockwise along an axis aligned with the momentum, will appear as rotation clockwise along that axis in a frame of reference that's overtaking the particle.

In contrast, no frame of reference exists that can overtake a light-speed particle. So, the clockwise versus counter-clockwise sense of the helicity is no longer relative, but is invariant.

The reason you'll see people say that "chirality is not helicity" is because it's not what's usually referred to as "spin". As I originally noted, for the helical luxons, there are only 3 pairs of conjugate coordinates, not 4. There is nothing that corresponds to the $(φ,p_φ)$ pair in the classical case; and in the quantum case, there is no "m" coordinate.

In fact, because there are only 3 conjugate pairs, it's actually possible to set it up in the same way as in the "spin 0" case, but only in a patchwork fashion. There is a fairly well-known result in the literature that says "photons can't be localized". What this refers to is that it is "impossible" to set up the conjugate pairs $(x,y,z)$ and $(p_x,p_y,p_z)$.

But it's only impossible in the same sense that it is "impossible" to set up two coordinates for a sphere. You can, but you have to do it in a patchwork fashion. I lay out the details here

Position Operator For Photons

that also contains the mathematical details of the above-mentioned symplectic Wigner classification for cases (1) and (2) ... as well as case (0), described therein, of the "homogenous" family, but not of case (3), also described therein, of the "tachyon" family.

Finally, back to your original question: where it not for the Higgs, all fundamental fermions would be light speed particles. As far as their interaction with the weak nuclear force is concerned, Higgs or not, they are all light-speed particles! So, the chirality is invariant. For matter, only left-handed particles have weak charge, and for anti-matter, only right-handed particles do.

The interaction with the Higgs toggles left and right modes. Remembering what was noted above about the relative nature of clock-wise and counter-clockwise sense of helicity; intuitively, it is as if the Higgs brings about a wild zig-zagging back and forth of the light-speed particles' linear momentum, so that their averaged-out trajectories are those of sub-light particles. Correspondingly, the Higgs gives the fundamental fermions the appearance of having mass, with the mass being proportional to the particles' "Higgs charge".

So, as far as interacting with the weak force, the fundamental fermions are light speed and have fixed chirality. But as far as interacting with the world at the scales we see it at, by virtue of the Higgs interaction, they have the appearance of sub-light particles with positive rest mass and a not-fixed and not-invariant helicity.