Quarks and hadrons - University of .Particles and Symmetries CHAPTER 7. QUARKS AND HADRONS 7.2 Hadrons

  • View

  • Download

Embed Size (px)

Text of Quarks and hadrons - University of .Particles and Symmetries CHAPTER 7. QUARKS AND HADRONS 7.2...

  • Chapter 7

    Quarks and hadrons

    Every atom has its ground state the lowest energy state of its electrons in the presence of theatomic nucleus as well as many excited states, which can decay to the ground state via theemission of photons. Nuclei composed of multiple protons and neutrons also have their ground stateplus various excited nuclear energy levels, which typically also decay via emission of photons (plus and radiation). But what about individual protons or neutrons?

    It was asserted earlier that individual nucleons are also composite objects, and may be viewed asbound states of quarks. And just as atoms and nuclei have excited states, so do individual nucleons.

    The force which binds quarks together into bound states is known as the strong interaction, and thetheory which describes strong interactions (on distance scales small compared to a fermi) is calledquantum chromodynamics, often abbreviated as QCD. The quarks carry a corresponding charge,the analogue of electric charge, which is labeled color charge, leading to the chromo in thename. Unlike electric charge, for which there is only a single variety - either plus or minus (withthe underlying symmetry group U(1)), there are three possible colors for the color charge of aquark, along with the corresponding anti-colors. The group describing the underlying symmetryis SU(3). 1 We will have more to say about QCD as we progress. But the justification for thevalidity of the following qualitative description of quarks and their bound states lies in the success ofQCD as a description of what is observed in the laboratory. Using this theory, one can do detailedquantitative calculations of the masses and other properties of bound states of quarks and comparewith experimental results. The theory works. (In fact, the story of how this theory has been verifiedin experiment, even though the theory has quarks as degrees of freedom, while experiment neverdetects individual quarks, is an interesting one indeed. We will have only a limited opportunity todiscuss it this quarter, but I will note that an essential feature of this story is the emergence of jetsof hadrons in the final states of high energy collisions, and I have spent much of my scientific careerclarifying both the theoretical and experimental properties of these jets.)

    In this Chapter we will introduce a variety of new concepts, which we will return to in more detailin subsequent discussion. In particular, we will be using several techniques from group theory. Nowwould be a good time to read both Chapters 10 and 11.

    1In the group theory language of Chapter 10 U(1) is an Abelian group and the particle corresponding to theinteraction, the photon, does not interact directly with itself, i.e., the photon has zero electric charge. SU(3), on theother hand, is a non-Abelian group and the gluons, the analogs of the photon, come in 8 varieties and do interactdirectly with each other, i.e., have nonzero color charge.

    c Stephen D. Ellis, 2014 1

  • Particles and Symmetries CHAPTER 7. QUARKS AND HADRONS

    7.1 Quark flavors

    Quarks are spin-1/2 particles (fermions), which come in various species, (playfully) referred to asflavors. Different quark flavors have been given somewhat whimsical names, as shown in Table 7.1(values from the PDG). Note that the table includes values for the quark masses (i.e., these arethe mass eigenstates), but care must be taken when interpreting these values as individual, isolatedquarks are never observed experimentally. On the other hand, it should be clear that the masses fordifferent flavors vary substantially, and the experimental impact of these differences can be measured.

    flavor symbol mass charge

    up u 2.3+0.70.5 MeV/c2 23 |e|down d 4.8+0.50.3 MeV/c2 13 |e|

    strange s 95 5 MeV/c2 13 |e|charm c 1.275 0.025 GeV/c2 23 |e|bottom b 4.18 0.03 GeV/c2 13 |e|

    top t 173.21 0.51 0.71 GeV/c2 (Fermilab) 23 |e|173.34 0.27 0.71 GeV/c2 (Fermilab + LHC)

    Table 7.1: Known quark flavors

    Along with quarks, there are, of course, also antiquarks, denoted u, d, s, etc., with the same massesbut opposite electric charge as their partner. (So, for example, the u antiquark has charge 2/3 andthe d has charge +1/3 - note the non-integer values.) As suggested above, quarks are distinguishedfrom leptons by an additional quantum number that is called color, which takes three possible values:say red, blue, or green (and anti-red, anti-blue and anti-green for the antiquarks). These names aresimply labels for different quantum states of the quark.2 Since quarks have spin 1/2, they can alsobe labeled by their spin projection, or , along any chosen spin quantization axis. Hence, for eachquark flavor, there are really six different types of quark, distinguished by the color (red, blue, green)and spin projection (up, down).

    In addition to the curious names, two other things in Table 7.1 should strike you as odd: the enormousdisparity of masses of different quarks, spanning five orders of magnitude, and the fact that quarkshave fractional charge (in units of |e|). Both issues are at the core of ongoing research, including thatat the LHC, seeking evidence of the dynamics of mass generation (now apparently at least partiallyclarified by the discovery of the Higgs boson) and the connection between quarks and leptons.

    2These names are purely conventional one could just as well label the different color states as 1, 2, and 3. Butthe historical choice of names explains why the theory of strong interactions is called quantum chromodynamics: aquantum theory of the dynamics of color although this color has nothing to do with human vision!


  • Particles and Symmetries CHAPTER 7. QUARKS AND HADRONS

    7.2 Hadrons

    No (reproducible) experiments have detected any evidence for free, i.e., isolated quarks. Moreover,there is no evidence for the existence of any isolated charged particle whose electric charge is notan integer multiple of the electron charge. This is referred to as charge quantization. Consistentwith these observational facts, the theory of strong interactions predicts that quarks will always betrapped inside bound states with other quarks and antiquarks, never separated from their brethrenby distances larger than about a fermi.3 The bound states produced by the strong interactions arecalled hadrons (hadros is Greek for strong).

    Quantum chromodynamics, in fact, predicts that only certain types of bound states of quarks canexist, namely those which are colorless. (This can be phrased in a mathematically precise fashionin terms of the symmetries of the theory. More on this later. For now consider this situation as beingsimilar to that in atomic physics where the low energy states, the atoms, are electrically neutral.)Recall that to make white light, one mixes together red, blue, and green light. Similarly, to make acolorless bound state of quarks one sticks together three quarks, one red, one blue, and one green.But this is not the only way. Just as antiquarks have electric charges which are opposite to theirpartner quarks, they also have opposite color: anti-red, anti-blue, or anti-green. So another wayto make a colorless bound state is to combine three antiquarks, one anti-red, one anti-blue, and oneanti-green. A final way to make a colorless bound state is to combine a quark and an antiquark,say red and anti-red, or better the truly colorless combination rr + bb + gg. Bound states of threequarks are called baryons, bound states of three antiquarks are called antibaryons, both of which arefermions, while quark-antiquark bound states are called mesons and are bosons.

    How these rules emerge from QCD will be described in a bit more detail later. For now, lets justlook at some of the consequences. The prescription that hadrons must be colorless bound states saysnothing about the flavors of the constituent quarks and antiquarks. In the language of quantummechanics we say that color dependent operators and flavor dependent operators commute,

    [color, flavor] = 0 . (7.2.1)

    Since quarks come in the multiple flavors of Table 7.1, we can (and will) enumerate the variouspossibilities for the hadrons. Similar comments apply to the spatial (orbital angular momentum)and spin parts of the wave function (i.e., the dynamics of these various parts commute to a goodapproximation), and we can think of the wave functions describing the hadrons, again to a goodapproximation, as products of a color wave function, a flavor wave function, a spatial wave functionand a spin wave function. The most important violation of this assumption of the factorization of thewave function arises due to the role of the Pauli Exclusion Principle for baryons, as we will shortlysee.

    For the lowest mass hadrons we may assume that the quarks are essentially at rest (except for theconstraints of quantum mechanics) and that the orbital angular momentum vanishes, ~L = 0. Thusthe rest energy of such a hadron (like any bound state, although it is admittedly a bit more subtle

    3Except at sufficiently high temperatures. Above a temperature of Tc 2 1012 K (or kT 170 MeV), hadronsmelt or vaporize and quarks are liberated. This is important in the physics of the early universe, since temperaturesare believed to have exceeded this value in the earliest moments after the big bang. Temperatures above Tc can alsobe produced, briefly, in heavy ion collisions. A nice overview of heavy ion collisions and quark gluon plasma may befound at www.bnl.gov/rhic/physics.asp . There is also an ongoing heavy ion program at the LHC, i.e., some runningtime is dedicated to accelerating and colliding heavy nuclei rather than protons.



  • Particles and Symmetries CHAPTER 7. QUARKS AND HADRONS

    in this case) may be regarded as the sum of the rest ene