LOG#180. Hadronic mysteries.

Hi, everyone!

Today is not today. Elementary particles are known to fall in two species, taking for granted the Standard Model (SM): bosons and fermions. The SM bosons are photons, gluons, W and Z bosons and the recently found Higgs boson. Bosons are “force” carriers (May the bosons be with you!), so photons carry electromagnetic forces, gluons carry chromodynamic forces, and W, Z carry weak forces (namely, they are responsible of flavor changing and ultimately of radioactive decay!). Moreover, at high energy, about or around hundreds of GeV, electromagnetic and weak forces are “unified” into electroweak forces. Fermions or leptons come into 3 families (only differing into mass, i.e., they are just one generation at common energies, but there are two more “families” and higher energies). Fermions can be leptons or hadrons. Leptons are the electron, the muon, the tau particle and their associated (massive, at least one state or flavor, due to neutrino oscillations). Leptons do NOT feel the strong nuclear force. However, hadrons do FEEL strong nuclear forces. Hadrons and their types are “complicated”. Hadrons are made of quarks (quarks and leptons make the fundamental fermions in the SM framework). But quarks, due to the asymptotic freedom of QCD (Quantum Chromodynamics) have the weird feature that can not be seen single! Quarks are never and never isolated. We deduce their existence due to phenomenological jets at colliders. Even worse, hadrons classification is puzzling. With the recent announcement of the \Xi_{cc}^{++} particle, I feel the need to make a little bit of quark chemistry, and to summarize some issues and enigmas not yet understood in the SM.

Let me begin this post naming the two more known baryons. You know them if you have basic education. Atoms are made of electrons AND the nucleus. The nuclei contain or are formed mainly (at least, the normal matter you know, you will know about other exotics in this post) by two particles or states. These states are baryons (baryons are made of 3 quarks; or even more odd number of quarks, read below…). Other states in the nuclei are mesons (quark + antiquark particles). Protons and neutrons are made of quarks as follows

(1)   \begin{equation*} p/p^+=N^+\equiv (u u d)=\mbox{proton} \end{equation*}

(2)   \begin{equation*} n=n^0/N^0\equiv (u d d)=\mbox{neutron} \end{equation*}

Sometimes, nuclear physicists call protons and neutrons as nucleons, and think about them as different STATES of the same entity, the NUCLEON. As quantum states, protons have a mass about 938 MeV/c², while neutrons a little bigger, 939.6 MeV/c². Both of them, are just about 1 GeV. The tiny difference of neutron and proton masses is far reaching. It is due to symmetry breaking and it is necessary in order life to exist (otherwise, atoms as we know would not exist!). You could imagine a world with neutrons lighter than protons, and it would be a weird universe. Baryons and mesons have to be colorless. So, the color charge, generally red (Red), green (G) and B (blue) has to be “white” inside any hadron (that is why mesons do form; you can glue red-antired, and so on, bound states). You can know the whole quantum numbers of quarks from the next table:

What else? Well, that is the question! We discovered that protons and neutrons ARE NOT alone. There are many others. Perhaps too many? Perhaps too few? You will think about this. Beyond mass, the “elementary particles” like hadrons are classified with two numbers (yep, do you like numbers?). The first one is the angular momentum in units of \hbar. It is denoted by J. The other number is called “parity”, and it is symbolized by P. Thus, a complete particle is listed within a J^P order. Of course, this is not even enough, and new “quantum” numbers must be introduced. But I am not going to go that far today. I am going to be simple. Using the J^P numerology, I am going to show you every hadron (baryon and meson) that the SM says it exist. Do you remember the quarks above? There are three generations, but, as leptons come in “pairs”, so there are 6 “flavored leptons” (do you like hexapods?), and so, you find experimentally 6 quark species (why? That is a good question! But again, not to be covered today…). The six quarks are named up (u), down (d), charm (c), strange, top or truth (t), bottom or beauty (b). You surely noticed neutrons and protons are made of up and down. Yes! And it was the discovery of new hadrons what hinted about the existence of quarks (at least, valence quarks are useful; fundamental quarks too).

Angular momentum for the simplest baryons is J=1/2. And parity comes in two types: plus (+) and minus (-). Plus parity is usually watched as “vector”-like, while minus parity is pseudovector-like for baryons (it would be scalar or pseudoscalar for mesons). We begin the baryonology zoo with the lambda particle:

\Lambda^0=(u d s) \Lambda^+_c=(u d c) \Lambda^0_b=(u d b)

These particles are three, and they are essentially udx states.

Surprised? You should not, not yet! There are many other baryons (net yet observed all of them!). Let me introduce you to the sigma particles, 9 states:

\Sigma^+=(uus) \Sigma^0=(uds) \Sigma^-=(dds)

\Sigma_{c}^{++}=(uuc) \Sigma_c^+=(udc) \Sigma^0_c=(ddc)

\Sigma_{b}^{+}=(uub) \Sigma_b^0=(udb) \Sigma_b^-=(ddb)

9 sigmas. Do you like the nonet, 3×3 array? Did you see it? Yes, certain sigma particles have the same quark composition that some lambda particles. How do we know they are different? Mass and other quantum numbers…Puzzled? Well, let me continue the tour.  Now, the awesome xi particle (that who triggered this post is listed here), sometimes dubbed cascade B or cascade particles:

\Xi^{0}=(uss) \Xi^-=(dss) \Xi_{c}^{+}=(usc) \Xi_{c}^{0}=(dsc) \Xi^{'+}_c=(usc) \Xi^{'0}_{c}=(dsc)

\Xi_{cc}^{++}=(ucc) \Xi_{cc}^{+}=(dcc) \Xi^{0}_{b}=(usb) \Xi_{b}^{-}=(dsb) \Xi^{'0}_{b}=(usb) \Xi_{b}^{'-}=(dsb)

\Xi^{0}_{bb}=(ubb) \Xi^{-}_{bb}=(dbb) \Xi^{+}_{cb}=(ucb) \Xi^{0}_{cb}=(dcb) \Xi^{'+}_{cb}=(ucb)  \Xi^{'0}_{cb}=(dcb)

So, you have 18 wonderful xi particles. Note that they are all made up from (uxx) or (dxx), with “x” being a quark of the second/third generation. Differences with the sigmas are quantum. Finally, the last hadron with J=1/2. The Omega baryon/particle.  These baryons come in 8 types:

\Omega_c^0=(ssc) \Omega_b^-=(ssb) \Omega_{cc}^+=(scc) \Omega^0_{cb}=(scb)

\Omega_{cbb}^0=(cbb) \Omega_{ccb}^+=(ccb) \Omega^{-}_{bb}=(sbb) \Omega^{'0}_{cb}=(scb)

8 omega particles! Do you have the \Omega? Do you watched the Saint Seiya Omega cartoon? Even, did you know Star Trek and the Borg veneration of the Omega particle? Do you know the omega Directive? 😉

Let’s count the above J=1/2^+ states: nucleons (2), deltas (3), sigmas (9), xis (18), omegas (8). Total: 40. Twice if you count the antiparticles. 80 particles plus antiparticles. Now, we proceed with the J=3/2^+ particles…Again, let me introduce the higher angular cousins of deltas, delta “resonances”:

\Delta^{++}=(uuu) \Delta^+=(uud) \Delta^0=(udd) \Delta^-=(ddd)

Curiously, we have here 4 (not 3!) delta particles. Again, we get the sigmas, now dubbed starred sigmas \Sigma^\star, to distinguish them from the previous sigmas.

\Sigma^{\star +}=(uus) \Sigma^{\star 0}=(uds) \Sigma^{\star -}=(dds)

\Sigma_{c}^{\star ++}=(uuc) \Sigma_c^{ \star +}=(udc) \Sigma^{\star 0}_c=(ddc)

\Sigma_{b}^{\star +}=(uub) \Sigma_b^{\star 0}=(udb) \Sigma_b^{\star -}=(ddb)

Now, we find out 12 starred xi particles (the primed versions are not known, but you can wondered if they should be allowed to exist as exercise):

\Xi^{\star 0}=(uss) \Xi^{\star -}=(dss) \Xi_{c}^{\star +}=(usc) \Xi_{c}^{\star 0}=(dsc)

\Xi_{cc}^{\star ++}=(ucc) \Xi_{cc}^{\star +}=(dcc) \Xi^{\star 0}_{b}=(usb) \Xi^{\star 0}_{bb}=(ubb)

\Xi^{\star +}_{cb}=(ucb) \Xi^{\star -}_{b}=(dsb) \Xi^{\star -}_{bb}=(dbb) \Xi^{\star 0}_{cb}=(dcb)

And now, the last \Omega particles. Now a decuplet (not an octet)! The fascinating “excited” (starred) omega particles are (2 unstarred as it is conventional):

\Omega^-=(sss) \Omega^{\star 0}_{c}=(ssc) \Omega_b^{\star -}=(ssb) \Omega^{\star +}_{cc}=(scc) \Omega_{cb}^{\star 0}=(scb)

\Omega_{bb}^{\star -}=(sbb) \Omega^{\star ++}_{ccc}=(ccc) \Omega_{ccb}^{\star +}=(ccb) \Omega_{cbb}^{\star 0}=(cbb) \Omega_{bbb}^{-}=(bbb)

Let’s count again the states: 4+9+12+10=35. Up to 70 if you count the antiparticles as well! Thus, we have no less than 40+35=75 particles, or 150 particles and antiparticles of baryonic nature. Note the unexisting (by the moment) of higher angular moment xi particles, and that there is no hadron with top quarks! That is simple. The top quark is so heavy, that it can not hadronize. Top quarks are not able to bound and form hadrons because they are too massive. A riddle: why is the top quark so massive with respect to the remaining quarks? Nobody knows for sure. Write me if you guess a good reason.

 

Are you ready for more? Let’s begin with meson listing. The first big group are the pseudoscalars with J^P=0^-. These are the pions (Yukawa intermediate effective strong force mediators!):

\pi^+=(u\overline{d}) with antiparticle \pi^-=(\overline{u}d)

\pi^0=\dfrac{1}{\sqrt{2}}=(u\overline{u}-d\overline{d})

You have 2 pions (3 if you consider the pion minus antiparticle). Then you have the 4 eta particles (their own antiparticles):

\eta=\dfrac{1}{\sqrt{6}}(u\overline{u}+d\overline{d}-2s\overline{s})

\eta^{'}=\dfrac{1}{\sqrt{3}}(u\overline{u}+d\overline{d}+s\overline{s})

\eta_c=(c\overline{c})

\eta_b=(b\overline{b})

Now, 4 kaons. The kaons are “special”. Due to CP violations, some of the following states are not really quite “accurate”. I mean, the K_S, K_L states are not exactly as I will write them, but only “approximately”. Yeah, it has become stranger. Stranger things happen in particle physics! States that are composite and a bit “fuzzy” due to CP-violations! Strange are kaons, thus:

K^+=(u\overline{s}) with antiparticle K^+=(\overline{u}s)

K^0=(d\overline{s}) with antiparticle K^0=(\overline{d}s)

K_S^0=\left(\dfrac{d\overline{s}+s\overline{d}}{\sqrt{2}}\right) equal to its antiparticle

K_L^0=\left(\dfrac{d\overline{s}-s\overline{d}}{\sqrt{2}}\right) equal to its antiparticle

Therefore, we get now 4 (6) kaons (kaons and antikaons). Next, the D-mesons and the B-mesons:

D^+=(c\overline{d}) with antiparticle D^-=(\overline{c}d)

D^0=(\overline{u}c) with antiparticle \overline{D}^0=(u\overline{c})

D^+_S=(c\overline{s}) with antiparticle D^-_S=(\overline{c}s)

B^+=(u\overline{b}) with antiparticle B^-=(\overline{u}b)

B^0=(d\overline{b}) with antiparticle \overline{B}^0=(\overline{d}b)

B^0_s=(s\overline{b}) with antiparticle \overline{B}^0=(\overline{s}b)

B^+_c=(c\overline{b}) with antiparticle B^-_c=(\overline{c}b)

So, 7 mesons (14 counting their antiparticles). In total, we have 7 (14)+ 4 (6) + 4 (4)+2 (3)=17 (27).

Finally, the vector mesons with J^P=1^-. Let me begin in the inverse way this time. The excited D-mesons and B-mesons are again 7 (14):

D^{\star +}=(c\overline{d}) with antiparticle D^{\star -}=(\overline{c}d)

D^{\star 0}=(\overline{u}c) with antiparticle \overline{D}^{\star 0}=(u\overline{c})

D^{\star +}_S=(c\overline{s}) with antiparticle D^{\star -}_S=(\overline{c}s)

B^{\star +}=(u\overline{b}) with antiparticle B^{\star -}=(\overline{u}b)

B^{\star 0}=(d\overline{b}) with antiparticle \overline{B}^{\star 0}=(\overline{d}b)

B^{\star 0}_s=(s\overline{b}) with antiparticle \overline{B}^{\star 0}=(\overline{s}b)

B^{\star +}_c=(c\overline{b}) with antiparticle B^{\star -}_c=(\overline{c}b)

You then get the rho particles 2 (2+1 antiparticle=3):

\rho^+=(u\overline{d}) with antiparticle \rho^-=(d\overline{u})

\rho^0=\left(\dfrac{1}{\sqrt{2}}(u\overline{u}-d\overline{d})\right)

Then, you obtain the four self-particles (particles = antiparticles):

\omega=\dfrac{1}{\sqrt{2}}(u\overline{u}+d\overline{d})

\phi=(s\overline{s})

J/\Psi=SP=(c\overline{c})

\Upsilon=(b\overline{b})

And finally, 2 excited kaons (plus their antiparticles, if you wish…2+2=4)

K^{\star +}=(u\overline{s}) with antiparticle K^{\star -}=(s\overline{u})

K^{\star 0}=(d\overline{s}) with antiparticle K^{\star 0}=(s\overline{d})

We sum now again for excited meson states: 2 (4) + 4 (4) + 2 (3) + 7 (14) = 13 (25). Together with previous results above 17 (27)=30 (52) particles. WoW. Anyway, for comparison, have a look the the following light meson table (it includes some resonances and states I gave up for simplicity):

 

You must be saying that it is over…Well, not so easy. I have only scheduled the main predicted (sometimes already confirmed as with the recent \Xi_{cc}^+ particle) particles. The hadron phenomenology is very complex and rich due to the subtle Yang-Mills theory it provides. Indeed, the existence of some of the above states came under surprise. Who ordered that? Said Rabi about the muon…But also with all these strange replicas…And more. Even even even worse. QCD calculations can indeed fit the hadron spectrum. The full match between experimental spectrum and the theoretical expectation coming from lattice QCD in supercomputers is far from clear. For instance, see this:

Furthermore, did you know that the mass-gap conjecture in the pure Yang-Mills theory is a Millenium Clay problem not yet solved? Did you know that experimentalists have found mischievous resonant particles of the listed particles I gave you? By resonance I mean that they are short-lived higher mass higher angular states of the “parent” particle (you can guess it with the prime particles and some baryons/mesons I listed). No one understand the inner degrees of freedoms of resonances or why they even exist for sure. In fact, lattice QCD suggests many other states, like the so-called glueball (hadron states made of purely gluon states), the exotics (HYBRIDS, hadrons states combining quarks with GLUONS in more complex manners), or even multi-quark species (tetra-quarks, penta-quarks, …),…So, the vacuum of QCD is not yet understood (just like its phenomenology, complex and hard, as you see). I mean, some people think that any baryon state could be written as follows:

(3)   \begin{equation*} \Psi=\vert \mbox{hadron}\rangle=\alpha_{3}\vert qqq\rangle +\beta_{5}\vert qqqq\overline{q}\rangle+\gamma\vert qqqg\rangle+\delta\vert ggg\rangle+\ldots \end{equation*}

What is an hadron (baryon, meson)? What are those resonances we see and their degrees of freedom? Where are the known missing “lost resonances” we expected and we did not found? What is the true nature of the QCD dynamics? One issue is how to understand some states/particles or resonances. Some people suggest that we could be seen the rise of quark chemistry. I have additional suggestive captures and pictures for you about this subject. Firstly, the debate

Secondly, the big molecular models of quarks that are yet to be decided

The issue of constituent vs. valence quarks, an old phenomenological discussion along the parton model by Feynman and others, and how to “image” not only the new multi-quark states (molecules?) but also the traditional and common nucleon states (what is the right “molecular geometry of the uud/proton and the udd/neutron quark states inside the hadron molecule? New studies suggest they are arranged like an “Y” instead a triangle…).

Perhaps quark-gluon chemistry is a better name, since indeed a big part of nucleons is due not the Higgs but the vacuum of the QCD! That is why the YM Clay problem matters. By the way, neutrons decay in about 888 seconds, but protons have been found to be very very stable. Current limitis are about 10^{33} years for proton lifetime! Hyper-Kamiokande will probe up to proton lifetimes about 10^{34-35}yrs. Lucky of us, of course, proton life-time is so big! But some GUTs and theories predict proton decay (bad news for future cosmic life-forms based on protons). Thus, the decay of formerly stable particles like the proton could be FINITE. We are protonically doomed. We or our descendants. Provided the Human Kind survives other cosmic catastrophes, we will have to face the proton decay rate in the far away cosmic future.

May the (strong) Force be with you! May the hadrons (QCD) and hyperons be with you!

P.S.: Some exercises for you!

Assignment 1. Any other insteresting hadron (baryon/meson) I did not include? Let me know!

Assignment 2. Search for the missing resonances I mentioned and what were/are the SM predictions not yet fully fullfilled in the quark/hadron espectrum. How many of them I did not include? Any others you found interesting?

Assignment 3. Why do particles DECAY after all? In addition, imagine a world or Universe where protons, neutrons or hadrons (or even any other SM particle, like heavy leptons) do NOT decay even being different entities/particles. What would it happen? Could it have been possible?

Assignment 4. Search for theories with quarks being non-fundamental. Pre-quarks? Pre-pre-quarks? Preons? Rishons? Why are they “complicated” and hard?

Assignment 5. Combinatorics with quarks is fun. Imagine we study quark chemistry from a purely mathematical viewpoint, and you gave up linear combinations for both mesons and baryons. You know there are 6 quark species or flavors. How many mesons can you make up from these quarks (consider a meson is a quark-antiquark couple)? How many baryons can you make up from these quarks? Do NOT count the corresponding antiparticles.

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