## Hypernuclei

A baryon is a particle made of 3 quarks. The most familiar are the proton, which consists of two up quarks and a down quark, and the neutron, made of two downs and an up. Baryons containing strange quarks were discovered later, since the strange quark is more massive and soon decays to an up or down quark. A hyperon is a baryon that contains one or more strange quarks, but none of the still more massive quarks.

The first hyperon be found was the Λ, or lambda baryon. It’s made of an up quark, a down quark and a strange quark. You can think of it as a ‘heavy neutron’ in which one down quark was replaced by a strange quark. The strange quark has the same charge as the down, so like the neutron the Λ is neutral.

The Λ baryon was discovered in October 1950 by V. D. Hopper and S. Biswas of the University of Melbourne: these particles produced naturally when cosmic rays hit the upper atmosphere, and they were detected in photographic emulsions flown in a balloon. Imagine discovering a new elementary particle using a balloon! Those were the good old days.

The Λ has a mean life of just 0.26 nanoseconds, but that’s actually a long time in this business. The strange quark can only decay using the weak force, which, as its name suggests, is weak—so this happens slowly compared to decays involving the electromagnetic or strong forces.

For comparison, the Δ+ baryon is made of two ups and a down, just like a proton, but it has spin 3/2 instead of spin 1/2. So, you can think of it as a ‘fast-spinning proton’. It decays very quickly via the strong force: it has a mean life of just 5.6 × 10-23 seconds! When you get used to things like this, a nanosecond seems like an eternity.

The unexpectedly long lifetime of the Λ and some other particles was considered ‘strange’, and this eventually led people to dream up a quantity called ‘strangeness’, which is not conserved, but only changed by the weak interaction, so that strange particles decay on time scales of roughly nanoseconds. In 1962 Murray Gell-Mann realized that strangeness is simply the number of strange quarks in a particle, minus the number of strange antiquarks.

So, what’s a ‘hypernucleus’?

A hypernucleus is nucleus containing one or more hyperons along with the usual protons and neutrons. Since nuclei are held together by the strong force, they do things on time scales of 10-23 seconds—so an extra hyperon, which lasts for many billion times longer, can be regarded as a stable particle of a new kind when you’re doing nuclear physics! It lets you build new kinds of nuclei.

One well-studied hypernucleus is the hypertriton. Remember, an ordinary triton consists of a proton and two neutrons: it’s the nucleus of tritium, the radioactive isotope of hydrogen used in hydrogen bombs, also known as hydrogen-3. To get a hypertriton, we replace one of the neutrons with a Λ. So, it consists of a proton, a neutron, and a Λ.

In a hypertriton, the Λ behaves almost like a free particle. So, the lifetime of a hypertriton should be almost the same as that of a Λ by itself. Remember, the lifetime of the Λ is 0.26 nanoseconds. The lifetime of the hypertriton is a bit less: 0.24 nanoseconds. Predicting this lifetime, and even measuring it accurately, has taken a lot of work:

Hypertriton lifetime puzzle nears resolution, CERN Courier, 20 December 2019.

Hypernuclei get more interesting when they have more protons and neutrons. In a nucleus the protons form ‘shells’: due to the Pauli exclusion principle, you can only put one proton in each state. The neutrons form their own shells. So the situation is a bit like chemistry, where the electrons form shells, but now you have two kinds of shells. For example in helium-4 we have two protons, one spin-up and one spin-down, in the lowest energy level, also known as the first shell—and also two neutrons in their lowest energy level.

If you add an extra neutron to your helium-4, to get helium-5, it has to occupy a higher energy level. But if you add a hyperon, since it’s different from both the proton and neutron, it can too can occupy the lowest energy level.

Indeed, no matter how big your nucleus is, if you add a hyperon it goes straight to the lowest energy level! You can roughly imagine it as falling straight to the center of the nucleus—though everything is quantum-mechanical, so these mental images have to be taken with a grain of salt.

One reason for studying hypernuclei is that in some neutron stars, the inner core may contain hyperons! The point is that by weaseling around the Pauli exclusion principle, we can get more particles in low-energy states, producing dense forms of nuclear matter that have less energy. But nobody knows if this ‘strange nuclear matter’ is really stable. So this is an active topic of research. Hypernuclei are one of the few ways to learn useful information about this using experiments in the lab.

For a lot more, try this:

• A. Gal, E. V. Hungerford and D. J. Millener, Strangeness in nuclear physics, Reviews of Modern Physics 88 (2016), 035004.

You can see some hyperons in the baryon octet, which consists of spin-1/2 baryons made of up, down and strange quarks:

and the baryon decuplet which consists of spin-3/2 baryons made of up, down and strange quarks:

In these charts I3 is proportional to the number of up quarks minus the number of down quarks, Q is the electric charge, and S is the strangeness.

Gell-Mann and other physicists realized that mathematically, both the baryon octet and the baryon decuplet are both irreducible representations of SU(3). But that’s another tale!

### 11 Responses to Hypernuclei

1. Imagine discovering a new elementary particle using a balloon! Those were the good old days.

The German word for cosmic rays is Höhenstrahlung, high-altitude radiation, though kosmische Strahlung, calquing the English term, is also used today. In the old days, one knew that it could be detected at high altitudes, but didn’t yet know its origin.

2. allenknutson says:

The central weight space in that baryon octet is 2-d, unlike all other weights spaces in the octet and the decuplet. What physical statements follow from that degeneracy?*

*guessed physicsese

• John Baez says:

Hmm, interesting. The people on Physics StackExchange are mainly asking about what is different about the two particles in that central weight space: the Λ, which is the hyperon I was talking about a lot already, and the Σ0. So I’ll answer that first, and then get to your actual question.

These two particles are similar in some ways. They’re both made of an up quark, a down quark and a strange quark, so they’re both electrically neutral. The Λ is 1.19 times as massive as the proton. The Σ0 is 1.27 times as massive.

On the other hand, they’re dramatically different in other ways!

The Λ can only decay via the weak interaction because it’s the lightest baryon containing a strange quark, and its also too light to decay into a proton and a strange meson: even the lightest strange meson is too heavy for that. So, the lifetime of the Λ is a whopping 0.26 nanoseconds.

The Σ0, being heavier, can decay into a Λ and a photon. This decay happens electromagnetically, I guess, so the lifetime of the Σ0 is much shorter: just 7.4 × 10-20 seconds!

How are they different mathematically? Well, you know that, but I’ll say it like a physicist.

The SU(3) group here acts on the $\mathbb{C}^3$ whose basis is u, d, and s (the three relevant quarks). It has an SU(2) subgroup that leaves s alone and acts on the $\mathbb{C}^2$ whose basis is u and d. This SU(2) symmetry is called isospin symmetry.

If we take the baryon octet and think of it as a representation of SU(2), it’s reducible. The Λ lives in its own little ‘isospin singlet’: that is, a 1-dimensional trivial representation of isospin SU(2). The Σ0, on the other hand, lives in an ‘isospin triplet’: that is, the 3-dimensional adjoint representation of isospin SU(2)

So, the Λ is a kind of loner, while the Σ0 is part of a little family containing also the positively charged Σ+ and the negatively charged Σ (which, of course, is not the antiparticle of the Σ+).

However, the Σ+ and the Σ cannot decay as quickly as the Σ0 can, because only the Σ0 has a lighter particle made of the same quarks: namely, the Λ. These other particles can only decay by getting rid of their strange quark! So, they decay via the weak interaction.

As it turns out, the Σ+ decays in 8 × 10-11 seconds, and the Σ decays in 1.5 × 10-10 seconds. These are respectably long amounts of time.

So, finally getting around to your question: what happens due to the Λ and Σ0 sitting at the same weight space in the baryon octet? The answer is that this enables the heavier one of these two particles to quickly decay into the lighter one!

Here are the mean lifetimes of the particles in the baryon octet. the Σ0 really stands out!

n: 879 seconds
p: infinite, as far as we know
Σ+: 8 × 10-11 seconds
Σ0: 7.4 × 10-20 seconds
Σ: 1.5 × 10-10 seconds
Λ: 2.7 × 10-10 seconds
Ξ0: 2.9 × 10-10 seconds
Ξ: 1.7 × 10-10 seconds

• We may as well cut out group theory. That is a subject that will never be of any use in physics.

— Sir James Jeans

• allenknutson says:

OK I think what I should be hearing here is, u/d/s have different masses so there is no real SU(3) symmetry. But if we pretend that they’re the same, then we have an SU(3) octet. Now we start to admit that the u/d isospin symmetry is a lot closer to being actually there than the SU(3) is. That decomposes the SU(3) rep under SU(2), splitting the central weight space into the 1-d trivial rep of SU(2) plus the central weight space of a 3-d rep of SU(2), and that’s the splitting into the hyperon and the other.

• John Baez says:

Allen wrote:

OK I think what I should be hearing here is, u/d/s have different masses so there is no real SU(3) symmetry. But if we pretend that they’re the same, then we have an SU(3) octet. Now we start to admit that the u/d isospin symmetry is a lot closer to being actually there than the SU(3) is.

Yes, all that is correct. The SU(2) symmetry between u and d is just approximate, and the SU(3) symmetry between u, d and s is even worse.

It’s a sign of divine benevolence that this approximate SU(3) symmetry led Gell–Mann and other physicists to take group representation theory seriously, eventually discovering the exact SU(3) × SU(2) × U(1) symmetry of the Standard Model, where the SU(3) does something completely different, and the SU(2) is somewhat related: the left-handed u and d quarks are a doublet for this “weak isospin”, but the right-handed ones are singlets.

In particle physics, especially in the early days, the approximate symmetry was easier to notice than an exact symmetry would have been: it’s often easier to see there are several suspiciously similar kinds of particles than to see there are several exactly identical kinds of particles.

Richard Block, a Lie theorist now emeritus at U.C. Riverside, regularly played tennis with Gell–Mann back when Gell–Mann was trying to understand baryons and mesons. Gell–Mann wanted a generalization of $\mathfrak{su}(2)$ to deal with strangeness. He tried 4-dimensional Lie algebras… 5-dimensional Lie algebras… 6-dimensional Lie algebras… 7-dimensional Lie algebras… and none of them worked! After tennis one day he mentioned this to Block, who suggested trying an 8-dimensional one: $\mathfrak{su}(3).$

3. Will says:

Minor correction: ” In 1962 Murray Gell-Mann realized that strangeness is simply the number of strange quarks in a particle, minus the number of strange antiquarks.”

Of course it’s just a convention, but strangeness is defined as the opposite (number of strange antiquarks minus number of strange quarks).

• John Baez says:

Wow! This reminds me of how Benjamin Franklin chose the sign of electrical charge so that electrons turned out to be negative… making the flow of electric current the opposite of the way the electrons are moving.

This chart on Wikipedia looks right:

but this one looks wrong:

4. Wyrd Smythe says:

Apologies if this is a dumb question, but if the Sigma-nought and the Lambda both are comprised of u-d-s quarks, what accounts for their mass difference?

• John Baez says:

Rather little of a baryon’s mass is due to the mass of its component quarks; most of the mass is due to the energy of the cloud of virtual quarks and gluons in the baryon. The u, d, and s are “arranged” differently in the Σ0 and the Λ, and this makes their cloud of virtual quarks and gluons have different energies.

What do I mean by “arranged”? Well, quantum theory is all about linear combinations of things, and very roughly, we have

$\Sigma^0 =[(ds-sd)u+(us-su)d]/2$

and

$\Lambda = [2(ud-du)s+(us-su)d+(sd-ds)u]/\sqrt{12}$

A computation showing this is here.

That’s a sketchy answer to a question that quickly gets very technical! It’s quite hard to compute the mass of a baryon from first principles, though it has been done.

• Wyrd Smythe says: