How powerful would a collider have to be to observe a sphaleron?

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In summary, at very high energies, interactions between particles violate the separate conservation of baryon number and lepton number. However, they still conserve baryon number minus lepton number.
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ohwilleke
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We haven't seen and as I understand it, won't see, sphaleron interactions at the LHC. What parameters would an experiment that could produce them need assuming that the Standard Model is valid up to arbitrarily high energies?
Sphaleron interactions are the only interactions in the Standard Model of Particle Physics that violate the separate conservation of baryon number and lepton number at very high energies, although they still conserve baryon number minus lepton number (B-L).

Even at the LHC we haven't actually observed a single sphaleron interaction, although it isn't impossible that this could happen in small numbers at some future collider (despite the nominal 13 TeV power of the LHC, it still doesn't get enough energy in a small enough place to produce sphaleron interactions in frequencies large enough to be detected over background events. But while I've seen the reasons for this briefly explained in passing by people more knowledgeable than I, these reasons are at the fringe of my ability to explain in technical detail).

In particularly, I don't understand the conditions necessary to give rise to a sphaleron interaction, or how distinctive a signature it would create if it occurred would be, well enough to be able to meaningful estimate what kind of collider or experiment (or "natural experiment" observed via astronomy for that matter) would be necessary to observe one assuming that the Standard Model is accurate up to arbitrarily high energies.

Some insight seems to be available in the body text of the article by N.S. Manton "The Inevitability of Sphalerons In Field Theory" (March 27, 2019) ("Based on Lecture at Royal Society Scientific Discussion Meeting: Topological Avatars of New Physics, 4-5 March 2019") at page 7-9 (citation omitted), which states that:

"The energy of the sphaleron depends on the Higgs boson mass and on the weak mixing angle. When the sphaleron solution was originally discovered, the Higgs boson had not been observed, and its mass was poorly constrained. The sphaleron energy was then estimated to be somewhere in the range 8 – 14 TeV. Now that the Higgs boson is known to have a mass of 125 GeV, a little more than the mass of the Z boson, the sphaleron energy is estimated to be approximately 9 TeV. This assumes that one can rely purely on the classical field equations, combined with the experimentally determined coupling and mass parameters. The contribution of the magnetic dipole field to the energy is only about 1%. . . . The sphaleron energy density is remarkably high. The length scale of the solution is the inverse of the masses of the contributing gauge and Higgs fields, of order (100 GeV)^−1 . This is approximately 10^−17 m, about 100 times smaller than the length scale of a proton. The sphaleron volume is therefore about 106 times smaller than that of a proton. As the sphaleron energy is about 104 times the mass of a proton, its energy density is about 1010 times that of a proton at rest. This, by itself, suggests the sphaleron is hard to produce.

Such energy densities appear to be unreachable in collisions at the LHC – CERN’s Large Hadron Collider. There, colliding protons each have an energy of more than 6 TeV (let’s optimistically call this 10 TeV) and they are Lorentz contracted in the centre of mass frame by a factor of 104 , the ratio of 10 TeV to the proton mass of 1 GeV. The energy density is therefore 108 times that of a proton at rest, and it is in the form of a rather thin pancake, as there is no transverse Lorentz contraction. This does not appear to be enough to produce sphalerons, although there are millions of collisions per second, and large fluctuations of the energy density must sometimes occur. Even if the energy density were two orders of magnitude larger, it could be hard to produce a sphaleron as the field energy, mainly in the form of quarks and gluons, would have to transfer into a coherent combination of W, Z and Higgs fields. Such a field can be interpreted as a coherent combination of about 10 each of W±, Z and Higgs particles. Therefore, the non-perturbative process of sphaleron production in particle collisions is generally thought to be exponentially suppressed, in the same way that soliton-antisoliton production is suppressed. However, the production rate may be enhanced if a strong magnetic field is present, in a region comparable to the sphaleron size. And the production rate is almost certainly enhanced at high temperatures.

Whether production of a sphaleron in particle collisions is at all likely may become clearer when experiments at LHC, or at somewhat higher energy, find evidence for simultaneous production of two or more Higgs particles together with a few W or Z bosons. The signal for this would be the production of several high-energy leptons (electrons, muons or neutrinos).

Remarkably, sphaleron production and decay is associated with a net change in baryon number B and lepton number L. This is the result of an anomaly in baryon and lepton number conservation laws, and related to the fact that a sphaleron has Chern–Simons number 1/2 . Sphaleron production is therefore potentially extremely important, as it may help us understand the baryon asymmetry of the universe. The universe is dominated by matter (protons) rather than antimatter (antiprotons) but the source of the asymmetry remains unknown (although there are many ideas). Certainly, any observation of baryon or lepton number violation would be revolutionary, as no experiment so far has ever detected such a violation. However, measuring a net change in baryon number in a high-energy collision may be hard, as many mesons and baryons, and also antibaryons, are produced. It may be easier to keep track of charged leptons, but the neutrinos carry lepton number too, and are generally undetected."


This helps, but it would be helpful if someone could better explain what kind of collider or experiment it takes to actually produce sphaleron energy of 9 TeV.

Would these be observable at one of the planned 100 TeVish next generation particle colliders? Would it matter if its was a lepton collider or hadron collider, or if it was linear or not? Would it need features not present in existing colliders to more tightly contain interaction energies in smaller volumes like some sort of magnetic "cage"? Would it take something more like a 1000 TeV collider of some kind? Or, it is something that simply isn't possible with foreseeable technologies due to some other limitations?

Pinning down the details of when sphaleron interactions occur experimentally would provide important grounding to cosmology theories of baryogenesis and leptogenesis, and it is one of the few phenomena predicted by the Standard Model since the beginning half a century ago or so, that like the Higgs boson prior to 2012, has not yet been observed. So, looking for this interaction could provide a strong physics justification for a more powerful future collider, if it was possible at a feasible energy scale, without the need to invoke speculative beyond the Standard Model physics models to justify such an immense expense.

To be clear, I'm not expecting a full technical answer at the advanced level regarding why this is or isn't the case, just some pointers in the direction of what conventional wisdom and accumulated knowledge is about what is necessary (recognizing that there isn't necessarily a definitive answer).

Also, I would presume that someone has figured this out already, but does the energy density of a Standard Model sphaleron start to get close to the energy density that would trigger a black hole singularity in general relativity? I could do that back of napkin calculation myself, but I fear that I'd get a unit conversion wrong or something like that and wouldn't trust my conclusion.

At an intuitive level, a normal stellar collapse black hole at just over neutron star mass has close to the mass-energy density of a proton at an order of magnitude level, but the necessary mass-energy density should rise a great deal at such tiny event horizon radii.
 
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It's not sufficient to have 9 TeV in total. You need that energy in the fields of the weak interaction.
The article you linked discusses this on page 8. It argues that even a 200 TeV collider is unlikely to be sufficient and then thinks about even higher energies with the comment
Whether production of a sphaleron in particle collisions is at all likely [...]

Lepton colliders would allow searching both for lepton and baryon number non-conservation in a relatively clean environment but I don't know if their collision can produce the necessary conditions for the sphaleron process at all. Plasma wakefield acceleration looks like the only method we have to think about such a collider.
Hadron colliders are limited to lepton numbers and even that would be difficult as it's easy to miss a low energy lepton. Even here we would probably need plasma wakefield acceleration. Try to make a giant circular accelerator and synchrotron radiation quenches your superconducting magnets. 500 TeV at 50 GeV/m would be "just" 10 km pure acceleration. Add more than that for focusing, add preaccelerators, accelerators for the wakefield production and so on and you end up with a giant complex.
 
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So, realistically, out of reach of a next generation collider, but perhaps another generation beyond that one purpose built for the task?
 

Related to How powerful would a collider have to be to observe a sphaleron?

1. How is a sphaleron related to a collider?

A sphaleron is a hypothetical particle that is predicted by the Standard Model of particle physics. It is believed that a collider would be needed to observe a sphaleron, as it is a high energy particle that can only be created in extreme conditions.

2. What is the purpose of observing a sphaleron?

Observing a sphaleron would provide evidence for the existence of the Higgs boson, which is a fundamental particle that gives mass to other particles. It would also help to further understand the mechanisms of particle interactions and the origins of the universe.

3. How powerful would a collider have to be to observe a sphaleron?

The exact power needed for a collider to observe a sphaleron is not known, as it depends on various factors such as the mass of the Higgs boson and the energy needed to create a sphaleron. However, it is estimated that a collider with energies in the range of 100 TeV (teraelectronvolts) would have a good chance of producing a sphaleron.

4. Are there any existing colliders that have the power to observe a sphaleron?

Currently, there are no existing colliders that have the power to observe a sphaleron. The Large Hadron Collider (LHC) at CERN, which is the most powerful collider in the world, has energies of up to 13 TeV and is not capable of producing a sphaleron. However, there are plans for future colliders, such as the proposed Future Circular Collider (FCC), which could potentially have the power to observe a sphaleron.

5. What are the challenges in building a collider powerful enough to observe a sphaleron?

Building a collider powerful enough to observe a sphaleron is a major technical and financial challenge. It would require the development of new technologies and materials to withstand the high energies and temperatures involved. It would also require a significant amount of funding and international collaboration. Additionally, there are safety concerns and ethical considerations that need to be addressed before such a collider can be built.

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