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Note: I refrain from using the concept of handedness and the terms left-handed and right-handed when referring to chirality since these usually refer to the helicity of charged fermions and their antiparticles and prefer to use instead the terms referring to their Lorentz invariant but not a constant of motion chirality, namely as Left-Chiral and Right-Chiral similar as looking statically at your two hands without any momentum involved. These handedness above terms sometimes used in common for either referring to helicity or chirality I find a big source of confusion even for the related WP pages references.

My question is:

It is stated in WP Chiral Theories,

"Particle physicists have only observed or inferred left-handed fermions and right-handed antifermions engaging in the charged weak interaction."

In the above quote WP uses the terms left-handed and right-handed referring to the chirality of these particles... ):

Since a right-chiral fermion would not carry any weak hypercharge thus it does not interact with the Higgs that would strongly imply that this electron does not gain any mass from the Higgs field. The same is also true for a left-chiral antifermion like a left-chiral positron (i.e. only right-chiral positrons carry a weak hypercharge).

But a massless electron cannot be described anymore as a physical electron.

Therefore I'm asking are there actually in nature any right-chiral isolated electrons or is this just a mathematical induced physics effective theory used in order to describe the Higgs mechanism?

Markoul11
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3 Answers3

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I would strongly disfavor your associating handedness with helicity instead of chirality, but no matter... Indeed, the WP usage of right-handed for right-chiral (annihilated by $1-\gamma_5$) is fine. It is a source of clarity, not confusion. In this issue, helicity stays out of the picture, regardless of language.

Since a right-chiral fermion would not carry any weak hypercharge thus it does not interact with the Higgs that would strongly imply that this electron does not gain any mass from the Higgs field....

Right-chiral fermion fields have a hypercharge proportional to their electric charge, by the (EWeak) Gell-Mann—Nishijima formula. They do interact with the Higgs field, in the Yukawa terms of vanishing total weak hypercharge. These terms couple right-chiral to left-chiral fermions and induce the fermion masses through the v.e.v. of the Higgs field, of course. (Charged current weak interactions refer to W couplings, and not Higgs couplings.)

Right-chiral electrons exist, and they are, on average, half the degrees of freedom my and your and the universe's electrons have.

They are never isolated, as the electron mass connects them to their left-chiral brothers. Chirality is not constant in time. They have little to do with the Higgs mechanism. (I hope you don't actually mean SSB, instead).

NB There is a pedagogical picture presenting this as a chirality oscillation controlled by the mass term in the hamiltonian. A stationary solution of the free Dirac equation dictates immediately, starting from the R state, $$ \langle \gamma_5 \rangle = \cos(2mt). $$

Cosmas Zachos
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  • https://tinyurl.com/29amj6fs quote: "The helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion and left-handed if opposite." end quote. https://tinyurl.com/3mayhbfu quote: "Particle physicists have only observed or inferred left-handed fermions and right-handed antifermions engaging in the charged weak interaction." end quote. Two WP pages using the same terms, left-handed and right-handed for describing different concepts namely helicity and chirality. Someone should tell them to fix this. – Markoul11 Mar 10 '22 at 22:22
  • Indeed, people using helicity sometimes slip into using the terminology of the chirality most associated with it at ultra-high energies; they would not be using helicity otherwise. This is taught and emphasized in all HEP courses. It is a convenient conflation, as long as it is not taken literally, as here. – Cosmas Zachos Mar 10 '22 at 22:30
  • Also to which the term "handedness" is referring to? Helicity or chirality or direction of rotation clockwise or counterclockwise makes confusion even worse. The PSE answer for example says that handedness is the helicity, https://physics.stackexchange.com/a/356788/183646 . Also this WP page, https://en.wikipedia.org/wiki/Chirality#Physics adds even more to the confusion. Is there a clear terminology standard defined in the literature? Any reference to be shared? – Markoul11 Mar 10 '22 at 22:41
  • Handedness is Chirality (from the Greek for hand, χείρ, as in "chiropractor"). For massless or almost infinitely energetic particles, positive helicity identifies with right-chirality, so people think of the two and refer to them interchangeably; for practical reasons. This happened and still happens for neutrinos, whose rest frame is hard to seriously contemplate. But a Lorentz boost can change helicity. It is a bit of a storm in a teacup, though... in the afternoon tea of pedantic theologians. You could ask a separate nomenclature question, but I hope you appreciate the facts. – Cosmas Zachos Mar 10 '22 at 22:54
  • @CosmasZachos Electroweak chirality is just one of many uses of “left-handed” and friends; it is neither the earliest, nor the most recent, nor the most common. People doing low-energy spin polarization talk intentionally about left- and right-handed helicities without qualification. Beware that left-helicity photons correspond to right-circularly-polarized electromagnetic fields; photons do not have chirality. Chemists talk about chiral molecules like dextrose and its reflection, which should be called “sinistrose” but which is disappointingly “L-glucose.” – rob Mar 11 '22 at 00:15
  • @rob yes; I'm happy with everything, as long as people firmly specify the facts involved... I won't attend the afternoon tea... I tend to use positive-negative helicity and L-R chirality, like most HEP people, fully aware of the breadth of usage among communities. In this question the issue is squarely on chirality, so I leave helicity alone... – Cosmas Zachos Mar 11 '22 at 01:38
  • @CosmasZachos Thanks for the interesting table link. – Markoul11 Mar 11 '22 at 08:55
  • @rob So, is it then helicity or chirality that is continiously flipping between left-handed and right-handed in an electron every time the electron "collides" with a Higgs Boson and the physical electron therefore gains its mass from the Higgs field? It is not sure if this page here with the term "handedness" refers to the helicity or chirality of the particles, https://bctp.berkeley.edu/neutrino/neutrino3.html (read 4th paragraph). – Markoul11 Mar 11 '22 at 09:05
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    I'll answer the above comment myself: every interaction with the Higgs Field, *not just the Higgs Boson, flips the chirality of the fermion, and the very same coupling with the Higgs field's v.e.v. is* the mass term in the lagrangian. Helicity loses much of its import in the presence of mass. – Cosmas Zachos Mar 11 '22 at 14:45
  • @CosmasZachos But an electron constantly alternating between left and right chiral simply means it is constantly flipping its intrinsic magnetic moment vector to the direction of its momentum or opposite to it, therefore also constantly flipping its spin vector which for a negative charge particle is always opposite to its magnetic vector. In simple words, the electron travels forward constantly flipping its N and S pole towards the direction of its momentum. It is the same electron. No need to describe it as two left and right chiral electrons in superposition https://tinyurl.com/2p87t7jf . – Markoul11 Mar 11 '22 at 19:21
  • Ach, no! You are still thinking about the helicity, which has no place in this question. The stationary electron does nothing of the sort! It has a spin that stays put. It is just the chirality that flip-flops with a frequency proportional to its mass... – Cosmas Zachos Mar 11 '22 at 19:29
  • Impossible. Unless it alternates continuously between an electron and a positron thus -e and +e. I cannot see any other explanation of changing its static chirality without changing its charge sign, unless the invention of the weak hypercharge comes in rescue... :) – Markoul11 Mar 11 '22 at 19:37
  • There is little more I could say, beyond inspection of the Dirac spinor…. – Cosmas Zachos Mar 11 '22 at 19:42
  • I think the previous gif says it all. On each gif the more fade blue helix lines in the background have opposite chirality than the ones on front. The two chiralities are alternating in time and are in superposition in space. I think I get it now. Thanks anyway for the discussion. – Markoul11 Mar 11 '22 at 19:51
  • I would point out, @CosmasZachos, that your "clarity" link says chirality is "only a label" while helicity is "physical handedness." Your discouragement of using "handedness" to refer to helicity states seems to run counter to this physical picture. – rob Mar 13 '22 at 02:49
  • Noted. This is obviously not what I would have written. The link should suffice though to completely exclude helicity from this specific discussion. – Cosmas Zachos Mar 13 '22 at 03:02
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Left-chiral and right-chiral electrons don't exist as isolated long-lived objects.

Instead, what we call an electron (with rest mass $m$), is actually a particle constantly oscillating between being a left-chiral electron ($e_L$) and a right-chiral electron ($e_R$). The frequency of this oscillation is extremely high, $\nu=\frac{mc^2}{h}=1.2\cdot 10^{20}\text{ Hz}$. Every time the electron changes from $e_L$ to $e_R$ or vice versa, it emits/absorbs a boson to/from the ubiquitous Higgs field which permeates the whole world.

enter image description here

Here I have paraphrased what I understood from Leonard Susskind's lecture "Demystifying the Higgs Boson" (especially in time 42:30 - 51:40).

Thomas Fritsch
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  • Are you sure it is not the helicity flipping sign but the chirality each time the electron collides with the Higgs Boson? Because I have seen the lecture from Susskind and I remember he was referring to the helicity and not the chirality, – Markoul11 Mar 10 '22 at 22:04
  • @Markoul11 May be you are right. I always struggle telling the difference between helicity, chirality and handedness. – Thomas Fritsch Mar 10 '22 at 22:09
  • Same also here. People use these terms interchangeable leaving the readers or audience guessing each time. This is very confusing. I suggest amending the terms left-handed and right handed always to which they are referring, helicity or chirality? For example, left-handed helicity or right-handed chirality or just right-chiral. Also avoid mixing the term "handedness" with helicity or chirality. It is my understanding that handedness refers to neither of these but instead to direction of spin rotation thus clockwise (left-handedness) or counterclockwise (right-handedness). – Markoul11 Mar 10 '22 at 22:55
  • @Markoul11 It’s usually clear from context whether you are referring to helicity, chirality, or the energy regime where they overlap. This is annoying for people who are coming into the field without much context. I am reminded of a place in Griffiths’ introductory quantum textbook where he uses $m$ twice in the same equation, in one place as a mass and in another place as the azimuthal quantum number. In a footnote he writes something like “yes, this is annoying. Everybody does it. Sorry.” – rob Mar 10 '22 at 23:33
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    @Markoul11 At 43... Susskind clearly is referring to chirality, since flips imply non constancy in time, which chirality "suffers", but not helicity; but, indeed, he is beginning his discussion by helicity, replete with mass gestures, in case you thought of spin as a spinning top, instead of a bland formal arrow. The leap between the two occurs when he drops the crucial condition of moving with a speed close to that of light. L.S. is famous for neat intuitive pictures with such hidden leaps. – Cosmas Zachos Mar 11 '22 at 15:09
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In its rest frame, an electron is an equal superposition of its left-chiral and right-chiral components. The two pieces transform differently under boosts, for reasons which occupy a chapter or two in textbooks on field theory. So if you are holding a polarized electron in your hand at rest, you’ll think it’s equal parts left- and right-chiral. But if I run past you relativistically, I’ll see your electron as one chirality or the other depending on whether I run at its “north pole” or its “south pole.”

The charged weak current only interacts with the left-chiral parts of the matter fields, and with the right-chiral parts of the antimatter fields. In the rest frame of the decay, electrons created by the charged current are purely left-chiral, and so the more energetic they are, the more strongly they are also left-polarized. But in the rest frame of the created electron, it is equal parts left- and right-chiral the whole time.

rob
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  • Does the left and right chiral phases for the electron at rest each lasts 2π of spin rotation? – Markoul11 Mar 10 '22 at 21:58
  • Are you referring to the rotational frequency of the electron’s intrinsic spin? There is no such frequency. The electron’s spin angular momentum is intrinsic to the field. – rob Mar 10 '22 at 22:05
  • This whole right left flipping reminds me to the Dirac Belt 4π spinor characteristic of the electron. Where each 2π the electron changes phase. Any correlation? – Markoul11 Mar 10 '22 at 23:02
  • A superposition between two states is different from “flipping back and forth,” even if there is some rotating phase associated with the superposition. Neither of these is related directly to the “double cover of the rotation group” that I think you are calling “Dirac’s Belt.” We rotate a lot of things in physics, in real space and in various kinds of phase space or configuration space. – rob Mar 10 '22 at 23:37