The Charged Sigma Baryons

The Field Guide to Particle Physics : Season 2
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The Charged Sigma Baryons

Introduction
The Sigma Baryons - that’s a capital Sigma - are a trio of slightly heavy cousins to everyday particles like the proton and the neutron.

With masses of almost 1200 MeV each, it may surprise you that the physics of Sigma baryons feels much closer to a comparatively puny trio of familiar particles: the pions.

The pions form a triplet of mesons: pi plus, pi zero and pi minus. So too, do the Sigmas: Sigma Plus, Sigma Zero and Sigma Minus. The similarities are helpful for building an intuition, but the differences are stark. While the charged pions are antiparticle partners, the charged Sigmas are anything but.

Today we’ll focus on that fact as we explore the pair of particles Sigma Plus and Sigma Minus.

Charged Sigmas

The charged Sigma baryons are your typical strange particle. They live much longer than they should, given their mass. Their lifetime is a sizable fraction of a nanosecond. Like the Lambda Zero, the charged Sigma baryons live so long because they have to wait for their constituent strange quark to decay.

The strange quark can only decay to an up quark, and while possible, it takes a while. It’s a quantum bottleneck that in particle decays that has come to be known as the technical term “Strangeness”.

While the down quark and the strange quark have separate identities as far as the strong nuclear force is concerned, they mix slightly under the weak nuclear force. That slight mixing is what gives the strange quark a chance to decay.

And it always decays to an up quark.

Keep an eye on this fact. It’s what makes the Sigma baryon decays so tricky.

What’s fun about the charged Sigma Baryons - that is markedly DIFFERENT than the charged pions - is that they are NOT antiparticles for one another. ¿The ANTI sigma plus is NOT the sigma minus. Not even close.

The Sigma Plus has two up quarks and a strange quark. That gives it’s electric charge of two thirds plus two thirds minus one third or one. The Sigma Minus has two down quarks and a strange quark, which contribute a charge minus one third each.

So, despite having opposite electric charges, they have very different quarks inside:up up and strange versus down down and strange.  And with that constitutional difference comes more mundane ones: the Sigma Plus and Sigma Minus have slightly different masses AND slightly different lifetimes. They are, in other words, very different particles.

Still. The Sigmas try their best to behave like pions. Isn’t it nice how neatly organized Nature at least tries to be?

At 1197 MeV, the Sigma minus is just a little bigger than the Sigma plus, whose mass is about 1189 MeV. Bigger masses usually imply short lifetimes, but the Sigma Baryons are strange in this sense too. The heavier, sigma minus baryon has a lifetime around 15% of a nanosecond. The lighter sigma plus baryon decays about twice as fast, living on average for about 8% of a nanosecond.

Charged Sigma Baryon Decays

Why does this slightly lighter, sigma plus baryon decay twice as fast?

Sigma Plus has two major ways to decay whereas Sigma Minus has only one.
Sigma Minus only really decays to a neutron and a pi minus. There are other options - including muons, electrons, neutrini and, rarely, a lambda zero-electron pair - which all together occur less than 1% of the time.

Similarly, 99% of the time Sigma Plus will decay into a familiar nucleon and a pion. But here’s a slight imbalance between these two options. The proton and pi zero appears just over 51% of the time. The neutron and pi plus happens a bit of 48% of the time.

Amusingly, the other 1% of stuff looks exactly like the anti particle versions of the rare Sigma minus decays. You know, antimuons, positrons and neutrinos. Notably, there’s also a rare Lambda zero with positron decay. Charge has to be conserved, after all.

Because the sigma plus has two ways to decay - two decay channels, in the parlance of particle physics - it’s not surprising that it decays twice as fast as its negatively charged sibling.

Why the sigma minus only has one decay channel relates back to the fact that is NOT the anti particle partner of sigma plus. Despite its negative charge, it’s made of QUARKS and not ANTIquarks. Because there is no negatively charged analog for the proton, there’s nothing else for the sigma minus to decay into.

Some Gory, Decay Details
The details of these decays are fun to examine.

The sigma minus - down, down, strange - decays when the strange quark does. The strange quark emits a W boson and leaves behind an up quark.  That essentially converts the Sigma minus into a neutron - down, down, up. The W boson promptly decays into a down quark-antiUp quark pair - that is, a negatively charged pion.

Did you get that? Sigma minus decays to a neutron with a pi minus.

The sigma plus - up, up, strange - is a bit more complicated. The strange quark again decays, but the final combination of quarks: up, up, up, down, anti-up, can be rearranged to form a proton: up, up, down and a neutral pion: up / anti-up. Because the W-boson lives for such a short time, that rearrangement all essentially happens at once.

The other possibility for the sigma plus is even more wild. The strange quark decay as usual, laving behind an up quark, but the emitted W-boson is immediately absorbed by one of the other up quarks, which then converts it into a down quark. If a gluon just happen to be emitted at around the same time, it can convert to a down quark-anti-down quark pair, giving a final combination of quarks: up, down, down, anti-down, up. This can be rearranged to form a neutron (up-down-down) and a pi plus (up-anti-down).

Was that complicated enough for you? Converting a sigma plus to a neutron is a little more complex so it doesn’t happen quite as often. To get a better sense of visualization, check out our drawing on the website. But suffice it to say, gluons aren’t hard to find given all that nuclear goo those quarks live with. It’s not all that surprising things work out this way.

We should say that these descriptions are something of a sketch or skeleton of what is actually going on. Physicists doing the full calculation using Quantum Field Theory would call it a tree-level approximation. Quantum effects can sometimes be dramatic, as we saw with the pi zero. Mercifully, not in this case.

Particle physics is nothing if not messy.

Why no Lambda Zero?

If you’re numerically minded - like you accountants out there - you might wonder why these charged Sigma baryons do not decay into a Lambda zero baryon. After all, the mass of the charged sigmas is around 1190 MeV, but the mass of the Lambda zero is only just shy of 1116 MeV.

Energetically, it’s more than possible! But the details matter.

Both Sigma plus and sigma minus can and do decay to Lambda 0 with either a positron or an electron, respectively.  But it’s a needle in a haystack. For every MILLION charged sigma baryons ...