LOG#225. QCD: the matrices.

Quantum Chromodynamics (QCD). The theory I learned that explains the nuclei as a teenager, i.e, the theory what explained why nuclei were not ripped off away due to nuclear forces. Strong nuclear forces (weak nuclear forces are the cause of radioactivity). Quarks and gluons. Matter and particle force carriers or messengers. I remember yet how some teens asked me and smiled with bad intentions what asked me what I was doing during the weekends and the week out of class. Sad news is that yet I have to keep calm seeing the same smile and similar comments about what I like to read, what I like to do or talk at my current age. Dark fate, I think…

Quarks are the matter fields feeling the strong nuclear force. Leptons do not feel color force. Color charges are just a name, not related to physical color though. It is just a way to work out a triad of gauge charges unlike electric charges (a diad) or gravitational charge (aka mass, a monad). How many colors are there? Three: red (R), blue (B) and green (G). Plus anticolor charges, antired \overline{R}, antiblue \overline{B} and antigreen \overline{G}. When two quarks interact, can swap charges or color via gluons. The interaction is something like a chemical reaction

    \[RB\leftrightarrow R\overline{B}\leftrightarrow BR\]

You can indeed imagine anticolor as color charges backwards in time. Expected gluons can be classified as follows (we will see that there are, in fact, only 8 possible color gluons independent to each other):

    \[\uparrow\uparrow=R\overline{R}\]

    \[\uparrow\downarrow=R\overline{B}\]

    \[\uparrow\downarrow=R\overline{G}\]

    \[\uparrow\uparrow=B\overline{R}\]

    \[\uparrow\uparrow=B\overline{B}\]

    \[\uparrow\downarrow=B\overline{G}\]

    \[\uparrow\downarrow=G\overline{R}\]

    \[\uparrow\downarrow=G\overline{B}\]

    \[\uparrow\uparrow=G\overline{G}\]

Note that these letters allow you to get colorless combinations with different quantum color numbers, thus being coherent with Pauli exclusion principle. States like uuu are possible ir you understand u_Ru_Bu_G, so quarks have different quantum numbers. Color is related as well to hypercharge and isospen, different combinations of isospin and hypercharge also label quarks and hadronic states. Experimentally, as color particles are not observable, only colorless states are possible. This fact settles a problem for B\overline{B},R\overline{R},G\overline{G} states, but, as surely you know, there are two main species of hadrons: baryons (quarky threesomes), and mesons (quarky couples). However, today hadron spectroscopy is much richer. Lattice field theories and experiments show us that there are also resonances AND, likely, exotics. Exotics are states not being purely couples or threesomes of quarks. You can get purely gluonic states called glueballs or gluonium by experts, and also you can get in principle quarky foursomes, quarky pentets and more. I read some ago a paper about heptaquarks and octoquarky states. Crazy! I wish I were such a quarky state sometimes.

What about gauge symmetries for quarks? It is called SU(3)_c symmetry. The fact protons get masses is due not to the Higgs mechanism, as I mentioned some days ago, but to a dynamical non-perturbative process of chiral symmetry breaking called sometimes dimensional transmutation. What a name! Gluons can be described by some 3\times 3 special unitary matrices or “grids”. These matrices have to be traceless (the sum over the main diagonal of its entries adds up to zero!). Let me represent color and anticolor states as column vectors and row vectors, respectively:

    \[R=\begin{pmatrix}1\\ 0\\0\end{pmatrix}\]

    \[B=\begin{pmatrix}0\\ 1\\0\end{pmatrix}\]

    \[G=\begin{pmatrix}0\\ 0\\1\end{pmatrix}\]

    \[\overline{R}=\begin{pmatrix}1& 0&0\end{pmatrix}\]

    \[\overline{B}=\begin{pmatrix}0&1&0\end{pmatrix}\]

    \[\overline{G}=\begin{pmatrix}0& 0&1\end{pmatrix}\]

Then, you can get by tensor product the 3\times 3 matrices

    \[R\overline{R}=R\otimes\overline{R}=\begin{pmatrix} 1&0&0\\0&0&0\\0&0&0\end{pmatrix}\]

    \[R\overline{G}=R\otimes\overline{G}=\begin{pmatrix} 0&1&0\\0&0&0\\0&0&0\end{pmatrix}\]

    \[R\overline{B}=R\otimes\overline{B}=\begin{pmatrix} 0&0&1\\0&0&0\\0&0&0\end{pmatrix}\]

    \[G\overline{R}=G\otimes\overline{R}=\begin{pmatrix} 0&0&0\\1&0&0\\0&0&0\end{pmatrix}\]

    \[G\overline{G}=G\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&1&0\\0&0&0\end{pmatrix}\]

    \[G\overline{B}=G\otimes\overline{B}=\begin{pmatrix} 0&0&0\\0&0&1\\0&0&0\end{pmatrix}\]

    \[B\overline{R}=B\otimes\overline{R}=\begin{pmatrix} 0&0&0\\0&0&0\\1&0&0\end{pmatrix}\]

    \[B\overline{G}=B\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&0&0\\0&1&0\end{pmatrix}\]

    \[G\overline{G}=G\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&0&0\\0&0&1\end{pmatrix}\]

There are 9 matrices, but R\overline{R},B\overline{B},G\overline{G} are not traceless. You get 6 matrices giving up these combinations. The SU(3) group has 8 generators (independent!). Where are the other 2? There are different choices, but you can that the following two matrices provide a good choice

    \[R\overline{R}-G\overline{R}=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}\]

    \[R\overline{R}+G\overline{G}-2B\overline{B}=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}\]

There are 8 total color grids/matrices for independent color changes:

    \[R\overline{G},R\overline{B},G\overline{R},B\overline{R},B\overline{G},G\overline{R},G\overline{B},R\overline{R}-G\overline{G},R\overline{R}+G\overline{G}-2B\overline{B}\]

Unlikely photons, gluons also interact with theirselves but not with the Higgs boson! Feynman graphs are typically Y shaped of double Y-shaped for gluon self-interactions. Cinfinement is a complex phenomenon. However, as said before here in TSOR, it can be modeled with a relatively simple potential for two quarks (the quarkonium):

    \[V(q\overline{q})=-\dfrac{4}{3}\dfrac{\alpha_s}{r}+kr\]

where k=\sigma is the string tension. This model is good for “heavy” quarks. This model can be compared to that of QED:

  • In QED, field lines extend up to infinity as \sim1/r^2. In QCD, quark field lines due to color are stretched or compressed into a tiny region between quarks and antiquarks. Breaking a flux tube requires the creation of a quark-antiquark pair.
  • In QED, Electromagnetic flux is conserved to infinity, but in QCD, color flux is trapped between quarks. No strong interations outside the color flux tube! Breaking the string implies a big energy, much larger than that of bounded atoms.
  • Hybrid states are allows, such as \langle q\overline{q}g\rangle, \langle ggg\rangle,\ldots. These exotica are the subject of some simulations of lattice QCD with supercomputers AND, as well, are being tried to mimic some of the resonant states already known. It is complicated to find out a good match, though.

In summary:

  1. QCD is all about quarks, gluons and internal symmetries modeled by matrices, matrices represent transitions between color states.
  2. QCD is the theory of color, the quantum number (not a physical color truly) of strong interactions. Have you ever imagined if we had called them strawberry, blueberry and mint? Or orange, vanilla and chocolate? Maybe, it would taste better…
  3. Gluons interact with theirselves, unlike photons, due to non abelian properties of the gauge group (something they share with weak and electroweak interactions).
  4. Gluons are Higgs transparent, they do not interact with the Higgs field. Why? Nobody knows. But it is fortunate for live to be it so.
  5. Mesons and baryons get masses due to the chiral symmetry breaking and a complex mechanism of dimensional transmutation. However, valence quarks can get a little mass from Higgs particles. So elementary quarks do interact with the Higgs field a little bit, but composite hadrons are getting masses through QCD.
  6. Hybrid states and resonances are expected to arise in the QCD spectrum. However, the Yang-Mills mass-gap problem is yet unsolved. Prove that tha Yang-Mills equations have a mass-gap and you will win a million dollar from Clay Institute!

See you in another blog post!!!!!

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