Exploring Angular Momentum in the Standard Model Higgs Condensate

[nnunn]

Thinking about the apparently rapid formation of galaxies in the early universe, is it possible for inhomogeneities to form, or even to exist, in a Higgs-type field?

If so, given the energy associated with such a (Higgs-type) condensate of weak hypercharge, could such inhomogeneities help to accelerate the formation of over-densities in the early universe?

 

[mitchell porter]

Topological defects, like monopoles, cosmic strings, and domain walls, are an example in which the vev of a condensate is not constant everywhere. And the idea does exist that the energy trapped in a cosmic string could gravitationally attract matter in the early universe, and thereby seed the formation of structure.

 

[nnunn]

"And the idea does exist that the energy trapped in a cosmic string could gravitationally attract matter in the early universe, and thereby seed the formation of structure."

Thanks Mitchell - interesting possible "causes" for possible non-homogeneity...

Before going on, with regard to interaction with a standard-model type of condensate of weak hypercharge, can someone clarify the difference between the sort of density disturbance the LHC can cause (and observe) vs. the sort of interaction (predicted by Brout, Englert and Higgs, Nobel 2013) that mediates the chiral oscillation of Dirac fermions?

 

[mitchell porter]

can someone clarify the difference between the sort of density disturbance the LHC can cause (and observe) vs. the sort of interaction (predicted by Brout, Englert and Higgs, Nobel 2013) that mediates the chiral oscillation of Dirac fermions?

Are you familiar with the relationship between particle and field in quantum field theory? i.e. the idea that particles are excitations of the field? A particle is really a quantum of energy in a field, distributed across the modes of the field in some way.

Owing to the self-interaction that defines the Higgs field's potential energy, the Higgs field has the property that even in vacuum it has a nonzero field strength, everywhere in space. That homogeneous and ubiquitous Higgs field strength is responsible for giving mass to the weak force bosons and the fundamental fermions. In the case of the fermions, you could argue that it changes the effective dynamics of the fermion fields, so that their excitations are massive four-component Dirac fermions, rather than massless two-component Weyl fermions.

In the case of the weak force bosons, it's not just that the dynamics of the weak force fields are affected, but particular particle excitations of the Higgs field now appear in coordination with the weak force fields. When massless, the weak force bosons (excitations of the weak force fields) have only two states of polarization, with positive and negative spin; but the "goldstone" excitations of the Higgs field act as de-facto third states of polarization, with spin zero.

Finally, there is a remaining particle excitation of the Higgs field which is, comparatively, an orphan. It doesn't line up with anything else in particular. That's what we call "the" Higgs boson.

So to sum up, there is the nonzero field strength taken on by the Higgs field as its lowest energy state, which then affects the dynamics of fields that are coupled to the Higgs field; and then, there are the particle excitations above this lowest energy state, one of which is "the" Higgs boson, the others de facto acting as weak force bosons with spin zero.

Now, to get topological defects in a field, that field has to have multiple possible lowest-energy states. Then, if the field stably settles into different minima in different regions of space - then you have a topological defect, formed by the boundary between those different regions.

The standard model Higgs field only has the one minimum. But with a more complicated set of fields, you can have multiple minima, and that's why you can have magnetic monopoles and cosmic strings in grand unified theories.

 

[nnunn]

Mitchell, thanks for sketching sound foundations.

Finally, there is a remaining particle excitation of the Higgs field which is, comparatively, an orphan. It doesn't line up with anything else in particular. That's what we call "the" Higgs boson.

Indeed. So assuming that this percussive (125 GeV) disturbance is not required to enable the chiral oscillation of fermions, or the oscillatory weak-hypercharging of Z-bosons, is it correct to associate such a "boson" with the mechanism predicted by Brout, Englert and Higgs?

As I understand it, the blip at 125 GeV is taken as evidence that there IS a condensate of weak hypercharge, and thus we take seriously a Higgs-type mechanism. Which points to the issue: what is it about a superposition of (massless) Weyl spinors that allows a Dirac fermion to absorb, and then to emit, quanta of weak hypercharge? How might a "quantized fluctuation" acquire, and then lose, such quanta?

Regarding non-homogeneity of this necessary condensate, the non-observation of domain walls expected if an "inflated cosmos" cooled through phase transitions got me wondering, not only about an inflated cosmos scenario, but also about "fluctuations" in "global fields".

Along these lines, a few years back (50th anniversary of electroweak unification), Steve Weinberg was asked if he was surprised by anything in the way particle physics had progressed. He replied that he was surprised we were "still using quantized fluctuations and wave equations." Not to imply that these tools were surprisingly good, but rather, that he had assumed some bright spark would have improved upon such place-holding machinery.

But it was two recent discussions that provoked this gazing beyond standard models:

First, near the end of this interview with Shaun Hotchkiss, discussing their July 2022 paper (v3) about the Pantheon+ supernova data set (arxiv.org/abs/2112.04510), Adam Riess and Dan Scolnic both agreed they'd run out of ideas for solving that famous discrepancy between expansion histories.

Then came Neil Turok's apology and candid admissions in this discussion with Brian Keating.

Given Neil's involvement in kick-starting and promoting inflated cosmos scenarios, etc., if he feels the need for a fresh start, I thought it may be worthwhile reconsidering my assumptions, too

 

[mitchell porter]

is it correct to associate such a "boson" with the mechanism predicted by Brout, Englert and Higgs?

The existence of that boson is a side effect of the Higgs mechanism (and unlike the mechanism per se, which was figured out by many people, Higgs was the only one to notice the leftover boson). The standard model Higgs field has four degrees of freedom, three of them synchronize with the W+, W-, Z fields, the fourth one is H, Higgs's boson.

what is it about a superposition of (massless) Weyl spinors that allows a Dirac fermion to absorb, and then to emit, quanta of weak hypercharge? How might a "quantized fluctuation" acquire, and then lose, such quanta?

We need to distinguish between (1) massless Weyl fermions being replaced by a massive Dirac fermion (2) either kind of fermion emitting or absorbing a Higgs boson.

In the Lagrangian of the standard model, there is a direct coupling between e.g. the electron field and the Higgs field. (2) is a consequence of that coupling; it means that energy can pass between the fields, which in particle terms corresponds to the creation and destruction of particles in interactions.

(1) is a consequence of that coupling, and of the nonzero Higgs field strength that exists even in vacuum. (This field strength is called a vacuum expectation value, or v.e.v., so I'll start using that term.) In other words, if the Higgs v.e.v. is zero, then the particles of the electron field are massless Weyl fermions; but if the Higgs v.e.v. is nonzero, then the particles of the electron field are massive Dirac fermions.

Mathematically, this is because the Higgs v.e.v. makes the electron-Higgs interaction term look like an electron mass term. I'm not sure how to explain it "physically", but it wouldn't surprise me if it could be explained as a form of entanglement between the Weyl modes of the electron field, mediated by their mutual coupling to the Higgs field.

Regarding non-homogeneity of this necessary condensate, the non-observation of domain walls expected if an "inflated cosmos" cooled through phase transitions got me wondering, not only about an inflated cosmos scenario, but also about "fluctuations" in "global fields".

OK, well, here we enter the true "beyond the standard models" realm, where there's no particular agreement about what's going on. :-)

Cosmology is complicated. I don't have a personally preferred explanation for the discrepancy in values of the Hubble constant, or even an opinion as to whether the "CMB dipole" is real or not. Meanwhile, at the start of this thread, maybe you were hinting at the too-early galaxies recently discovered by the James Webb space telescope. Since I am interested in MOND approaches to dark matter, it is interesting to me that a MOND paper also predicted those early galaxies. But there may be other ways to explain them.

Regarding inflation, there are umpteen different theories of what the inflation-causing field could be, but my favorites are Higgs inflation models, in which inflation is due to the Higgs field too! It's an attractively economical explanation, and it's another obscure hint at the significance of the Higgs boson mass (e.g. as if the size of the universe, the duration of inflation, and the Higgs mass are all somehow connected anthropically), but it gets bogged down in quantum gravity details that no one agrees on.

I think you started out wondering if inhomogeneities in the Higgs field could somehow seed structure and thereby produce those too-early galaxies. I mentioned the standard idea of topological defects, but that would require a Higgs sector more complicated than the standard model's minimal four-component Higgs. In my own ignorance, I nonetheless find it conceivable that there are other, more subtle phenomena implicit in the standard model that could do it. Perhaps there could be virtual topological defects (maybe the "electroweak monopoles" mentioned in a number of papers that I've never read).

Also, a fellow called Axel Maas has written a whole series of papers about a more fundamental approach to the Higgs mechanism, which he says implies subtle side effects. Maas isn't proposing any new fields or even a different interaction between them, instead he's saying that the usual picture of the standard model Higgs mechanism is a simplified approximation to how it really works. (His latest paper was just last week.) I have no idea if his approach could have consequences in the early universe, but theoretically it seems well-founded, so, it might be worth consideration.

 

[nnunn]

Mathematically, this is because the Higgs v.e.v. makes the electron-Higgs interaction term look like an electron mass term. I'm not sure how to explain it "physically", but it wouldn't surprise me if it could be explained as a form of entanglement between the Weyl modes of the electron field, mediated by their mutual coupling to the Higgs field.

Thanks Mitchell - that DOES sharpen the focus! Following this lead and shifting into the local frame of a superposition of entangled spinors, a necessary precursor to Weyl modes would seem to be Planck's quantum of angular momentum. But given (1) the way quantized angular momentum is connected with quantum spin, and (2) the pervasion of space with weak hypercharge, does standard model math allow Weyl modes to be "caused" by quanta of angular momentum acting on that condensate of primitive "pre-charge"?

With regard to tiny things interacting, back in 1961 Feynman was asked how he'd answer his own "cataclysmic question": "If scientific infrastructure were to be wiped out, what one scientific fact, that could be expressed in a few words, would he try to pass on?" He pointed to "atomicity", the idea that "all things are made out of atoms, little particles that move around, are in perpetual motion, attract each other when they're some distance apart but repel being squeezed into one another."

Decades later, having demonstrated a fractional quantum Hall effect, Bob Laughlin played with this idea of "atomicity". Faced with his data about fractional electric charge, and the fractional charges of quarks, and the range of masses for otherwise identical things (electron, muon, tau), and the idea that electric charge can be modeled as $Q = T_3 + \frac 1 2 Y_W$, he imagined arrangements of (quote) "little ghosts" linking arms (Robert Laughlin 2005, "A Different Universe: Reinventing Physics from the Bottom Down" page 42). Thinking about the non-trivial characteristics of standard model particles, rather than hope for perfect and identical particle-sized "fluctuations", he speculated that, from an engineering point of view, it would be easier to build up such identical and complex interactors from "piles" of smaller things (behaving like Feynman's uncuttable a-toms).

Along these lines, given that electroweak theory identifies and models so effectively the degrees of freedom bound up in Dirac's electron, can we take this model more literally? Would another level of quark-like confinement -- a next level of atomicity -- help with harnessing these spinors?

Mitchell, thanks again for adding clarity and insight.

 

[mitchell porter]

a necessary precursor to Weyl modes would seem to be Planck's quantum of angular momentum. But given (1) the way quantized angular momentum is connected with quantum spin, and (2) the pervasion of space with weak hypercharge, does standard model math allow Weyl modes to be "caused" by quanta of angular momentum acting on that condensate of primitive "pre-charge"?

Slightly rephrasing what I said in #6, I might put it like this. There are two fundamental causal interactions at work here, the self-interaction of the Higgs field, and the three-way "yukawa coupling" between two Weyl fermion fields and the Higgs field. The self-interaction of the Higgs field causes it to "condense" and form the condensate. This hypercharged condensate in turn causes the two Weyl fields to become entangled, via the yukawa coupling, forming a Dirac field.

My terminology of "Weyl mode" is not quite standard, so I will clarify that I mean "oscillatory mode of a quantum field, whose excitations are Weyl fermions". Also, I did say that "A particle is really a quantum of energy in a field", but particles do inherently carry angular momentum as well as energy-momentum.

The ontology of quantum fields is a little complicated because you don't just have oscillating quantities, as with the properties of a classical field; but those quantitative properties are subject to the uncertainty principle and to entanglement.

Would another level of quark-like confinement -- a next level of atomicity -- help with harnessing these spinors?

One does not need extra fields or more fundamental fields, in order to produce the standard model Higgs mechanism of electroweak symmetry breaking and mass generation. All that is needed, are the interactions I mentioned.

That said, I will mention some possible nuances. Axel Maas, who I also mentioned in #6, argues that the conventional picture of particles cleanly associated with distinct fields is just an approximation, and that in reality one needs to think of e.g. a physical electron as an excitation of the electron field and an excitation of the Higgs field, bound together.

Abbott and Farhi suggested something similar long ago, except that in their model, the two excitations are not just bound, but actually confined, in the same sense as QCD. However, I think this leads to subtle differences in the wavefunction structure of the electroweak bosons, that have been ruled out.

All these are just technical nuances of the standard model (Abbott and Farhi were also using the standard model, just with a large coupling for the weak force). Then for other topics we have discussed, like Higgs inflation, or the effect of inhomogeneities in the Higgs condensate on galactic structure formation, one needs to consider the standard model coupled to gravity, which brings new technical nuances.

Then there are the endless possibilities if one wants to actually change the standard model, with extra fields or more fundamental fields. I don't have Laughlin's book so I don't know what specific ideas he's promoting. In the work for which he's most famous, Laughlin obtained fractionally charged quasiparticles (for the quantum Hall effect), as collective excitations of a "spin liquid" of electrons; and possibly he wants to obtain quarks, which are also fractionally charged (relative to a scaling which assigns unit charge to electrons and protons), in a similar way.

The trouble with such schemes, and with preon (subquark) models in general, is that they usually also lead to extra wrong predictions, e.g. extra composite particles (other combinations of the subquarks) that should be observed, but aren't. Also, there is a difficulty in producing a light particle like an electron or a quark, by binding preons in the same way that quarks are bound in the proton; the binding energy of the preons needs to be large, so the composite particles will be heavy unless delicate cancellations occur. There is a loophole, a real example of which is given by pions. A pion is usually described as a quark-antiquark composite, but in fact it is actually an excitation of a quark-antiquark condensate which, like the Higgs condensate, fills the vacuum. A particle of this kind (a pseudo Goldstone boson) can be light.

Anyway, elite opinion is somewhat against preons, but large numbers of preon and related models have been proposed anyway. Whether one takes them seriously, depends on which hints for beyond-standard-model physics, one takes seriously. There is, for example, a class of models in which the Higgs field is actually made of top quark and top antiquark fields; so the Higgs would be similar to the pion, but the binding interaction would need to be something other than the gluon field. PF's @arivero has a paper which I regard as minor evidence for this. Another cute idea of his, a spinoff of his "sBootstrap" concept, is that maybe the muon is a kind of supersymmetric partner of the pion, a pseudo Goldstone fermion, and that this explains why the muon and pion have similar masses (similar enough that the muon was mistaken for the pion when it was first detected in cosmic rays). Then there's the Koide formula for electron, muon, and tau masses, popular here, which Koide originally stumbled upon in a subquark model. There was the recent thread on the Kahana model... There are a lot of ideas out there, about extra substructure or alternative ways to analyze structure.

 

[nnunn]

Mitchell, thanks for another such thoughtful and helpful reply. Before getting back to the issue of non-homogeneity in the distribution of weak hypercharge, can you clarify something about the way the standard model currently models interaction with this distribution?

Back in message #4 above, you wrote:

Finally, there is a remaining particle excitation of the Higgs field which is, comparatively, an orphan. It doesn't line up with anything else in particular. That's what we call "the" Higgs boson.

And in #6 you wrote:

We need to distinguish between (1) massless Weyl fermions being replaced by a massive Dirac fermion (2) either kind of fermion emitting or absorbing a Higgs boson.

Returning to my question back in #5:

(A)ssuming that this percussive (125 GeV) disturbance is not required to enable the chiral oscillation of fermions, or the oscillatory weak-hypercharging of Z-bosons, is it correct to associate such a "boson" with the mechanism predicted by Brout, Englert and Higgs?

It's precisely this emission and absorption of quanta of weak hypercharge that got me interested.

From what you describe, there does seem to be a distinction between (a) that which mediates the mass-inducing chiral oscillation of electrons, (b) the oscillatory weak-hypercharging of Z-bosons, and (c) your "orphan" excitation. So can you clarify what you mean by either kind of fermion "emitting or absorbing a Higgs boson" ?

When Susskind discusses this in his "Demystifying the Higgs" video (from time 48:10), he very carefully uses a place-holder "ziggs" particle to mediate both the oscillatory weak-hypercharging of Z-bosons, and the mass-inducing chiral oscillation of electrons. And like you, he later describes a Higgs-type boson as very much an "orphan excitation".

Regarding your comments on "preon (subquark) models" usually leading "to extra wrong predictions": Given the way the standard model builds up Lagrangian interaction from entangled superpositions of Weyl spinors, can the quantized action (angular momentum) locked into such (topologically protected?) spinors serve as a more fundamental uncuttable (a-tom) for Feynman's notion of "a-tomicity" mentioned in message #7 above?

Finally, getting back to the issue of non-homogeneity in the distribution of weak hypercharge: does current electroweak theory allow for any connection between (i) a Higgs type condensate and (ii) the electrical permittivity and magnetic permeability of space?

 

[mitchell porter]

I think the simplest way to proceed is to build on something @Haelfix wrote back in 2014.

We are going to be talking about standard model fields under two conditions. One is at low temperatures where electroweak symmetry is broken. The other is at temperatures above the "electroweak phase transition".

Haelfix mentions the Higgs field "H", and the electroweak gauge fields "W" and "B". To this I will add left-handed and right-handed Weyl fermions. The left-handed Weyl fermions come in pairs, like "left-handed electron" and "left-handed electron-neutrino", or "left-handed up quark" and "left-handed down quark".

Above the electroweak phase transition, all particles associated with these fields are massless, and the Higgs field has not "condensed". All the Weyl fermions can emit/absorb H, W, and B quanta... These conditions exist only in an ultra-high-temperature plasma, far hotter than any star or collider, such as in the very early universe.

At temperatures below the phase transition, the Higgs field is peaceful enough that it can "condense". One of its four real-number components gains a permanent field strength. Here I refer to Haelfix's exposition. He labels the four components as H+, H-, H0, h. h is "the" Higgs boson and corresponds to a particle excitation of the field component that gets a v.e.v. The other three excitations become part of the W and B fields, leaving us with massive W+, W-, Z0 particles, and the massless photon, γ.

As for the Weyl fermions, thank to the yukawa couplings among the Higgs field and the fermion fields, they have been "replaced" by Dirac fermions, in that henceforth, the "left-handed electron" and the "right-handed electron" only occur as possible states of the Dirac electron, etc. And meanwhile, "Weyl fermions can emit/absorb H, W, and B quanta" has become "Dirac fermions can emit/absorb h, W+/W-/Z0, and γ".

Is this making sense? The Higgs field is prevented from condensing if there's a particle plasma of ultra-high energy. But once the Higgs condensate can form, the kinds of particles that can form are different - massive Dirac fermions rather than massless Weyl fermions; three massive gauge bosons and a massive Higgs boson, rather than three massless gauge bosons and four Higgs excitations which are (I think) primordially massless. Only the photon is a remnant of the pre-condensate world, being the sole component of the W-B fields which is unaffected by the condensed component of the Higgs field.

 

[nnunn]

Mitchell, your post #10 above really helped to clarify relationships between Weyl spinors, Dirac spinors and chiral states of standard model electrons. Thanks for taking the time!

And thanks also for pointing to Haelfix's description of linear combinations of (non-Higgs) excitations of a Higgs-type condensate:

I think the simplest way to proceed is to build on something @Haelfix wrote back in 2014.

This led deep into standard model foundations, and got me wondering about another fundamental:

Given the standard model's dependence on [1] quantized action (Planck's constant), and [2] a condensate of weak hypercharge (Higgs-type field), do you know of any work that specifically explores the action of this quantized action on this condensate?

That is to say, how do theorists model a quantum of angular momentum acting on a superfluid condensate of weak hypercharge?

 

[mitchell porter]

how do theorists model a quantum of angular momentum acting on a superfluid condensate of weak hypercharge?

The standard model Higgs condensate is not a fluid - nothing about it is flowing - it's just sitting there. Also, it's a little misleading to call it a "condensate of weak hypercharge", as if hypercharge is a substance of which the condensate is made. It's a condensate of Higgs field, and the Higgs field has hypercharge; but hypercharge is a property possessed by most of the standard model fields. For example, quarks have hypercharge, and in fact the standard model vacuum also contains a hypercharged pion condensate (a pion being a quark-antiquark excitation) which by itself would give the electroweak bosons a small mass, in the absence of the Higgs field. (One of the historic theories of the Higgs field is that it is a kind of top-antitop "pion" held together by a new force.)

As for how a quantum of angular momentum would act on the Higgs condensate, it can't. The Higgs field is a spin-0 scalar field whose excitations do not have angular momentum. I guess it would be different if the Higgs field really was a superfluid, since a superfluid can have vortices.

Now, you may have run across explanations of how the Higgs gives mass to fermions, in which the fermion is depicted as flipping between left-handed and right-handed states while emitting Higgs bosons into the condensate. This may sound as if the Higgs interaction is changing the angular momentum after all. But you may find here an old post which describes this phenomenon as "mixing" between left-handed and right-handed states. As far as I can tell, it's just a kind of superposition. But I do need to study and think about this a bit further.